Erdős Problems

The Erdős problems are a large collection of questions posed or promoted by Paul Erdős and his collaborators (Bloom 2026). They range across number theory, combinatorics, graph theory, geometry, and related areas. Bloom's online Erdős Problems database listed 1221 numbered problems as of Oct. 26, 2026, together with their sources, status, references, and discussion. The numbers are database identifiers rather than established names for the individual questions.

The following table lists all 1221 problem numbers, using standard names where available and brief descriptions otherwise. Links identify existing coverage of the named problem or mathematical topic (Bloom 2026).

problem	name or description
1	Erdős distinct subset sums problem
2	minimum modulus of a covering system
3	Erdős-Turán conjecture
4	large gaps between consecutive primes
5	limit points of normalized prime gaps
6	three successively increasing prime gaps
7	distinct covering systems with odd moduli
8	covering systems with monochromatic moduli
9	odd integers not expressible as a prime and two powers of 2
10	representations as a prime and boundedly many powers of 2
11	odd integers as a squarefree number and a power of 2
12	infinite sets avoiding divisors of sums of larger elements
13	finite sets avoiding divisors of sums of larger elements
14	integers with nonunique or missing two-term sum representations
15	convergence of an alternating prime series
16	odd integers outside prime-plus-power-of-2 representations
17	even differences of primes below a prime bound
18	number of divisors needed in practical-number representations
19	Erdős-Faber-Lovász conjecture
20	sunflower conjecture
21	small intersecting families with large covering number
22	Ramsey-Turán problem for 4-cliques
23	making triangle-free graphs bipartite by edge deletion
24	pentagon counts in triangle-free graphs
25	logarithmic density outside truncated residue classes
26	divisors drawn from a translate of an infinite set
27	almost covering systems with comparable moduli
28	Erdős-Turán additive basis conjecture
29	explicit additive bases with few representations
30	error term for the largest finite Sidon set
31	additive complements of density zero
32	sparse additive complements of the primes
33	minimum growth of additive complements of the squares
34	distinct consecutive sums in permutations
35	Schnirelmann density growth under addition of a basis
36	minimum overlap problem
37	lacunary essential components for Schnirelmann density
38	density-increasing shifts from a set that is not a basis
39	dense infinite Sidon sets
40	density thresholds for unbounded additive representation counts
41	density of sets with distinct three-term sums
42	Sidon sets with disjoint nonzero difference sets
43	sizes of two Sidon sets with disjoint nonzero differences
44	extending finite Sidon sets to large Sidon sets
45	monochromatic Egyptian fractions using divisors of one integer
46	monochromatic Egyptian fraction representations of 1
47	Egyptian fractions in sets with large reciprocal sums
48	common values of the totient and divisor-sum functions
49	large sets on which the totient function increases
50	derivatives of the totient-ratio distribution function
51	large least preimages of totient values
52	sum-product problem
53	many subset sums or subset products
54	growth of Ramsey 2-complete sets
55	growth of Ramsey complete sets for more colors
56	largest sets without many pairwise coprime elements
57	reciprocal sums of odd cycle lengths in infinitely chromatic graphs
58	chromatic bounds from the number of odd cycle lengths
59	counting forbidden-subgraph graphs (four-cycle case: square-free graph)
60	4-cycle counts just above the extremal threshold
61	Erdős-Hajnal conjecture
62	common high-chromatic subgraphs of uncountably chromatic graphs
63	power-of-2 cycle lengths in infinitely chromatic graphs
64	Erdős-Gyárfás conjecture
65	reciprocal sums of cycle lengths in dense graphs
66	logarithmic asymptotics for additive representation counts
67	Erdős discrepancy problem
68	irrationality of the reciprocal factorial-minus-one series
69	irrationality of the binary prime-factor-count series
70	partition relations for triples at the continuum
71	cycles with lengths in a prescribed arithmetic progression
72	zero-density sets meeting cycle-length sets of dense graphs
73	near-bipartiteness forced by large independent sets in subgraphs
74	infinitely chromatic graphs with nearly bipartite finite subgraphs
75	large independent sets in finite subgraphs of uncountably chromatic graphs
76	edge-disjoint monochromatic triangles in two-colored complete graphs
77	exponential growth rate of diagonal Ramsey numbers
78	explicit exponential lower bounds for Ramsey numbers
79	minimal obstructions to Ramsey size linearity
80	large books in dense graphs whose edges lie in triangles
81	clique partitions of chordal graphs
82	large regular induced subgraphs
83	bounds for doubly intersecting uniform set families
84	number of possible sets of graph cycle lengths
85	monotonicity of the minimum-degree threshold for 4-cycles
86	4-cycle-free subgraphs of hypercube graphs
87	Ramsey lower bounds in terms of chromatic number
88	induced subgraphs with prescribed edge counts in Ramsey graphs
89	Erdős distinct distances problem
90	Erdős unit distance problem
91	nonuniqueness of point sets minimizing distinct distances
92	many equal distances from every point
93	distinct distances in convex polygons
94	second moments of distance multiplicities in convex polygons
95	second moments of planar distance multiplicities
96	unit distances in convex polygons
97	vertices with small distance multiplicities in convex polygons
98	distinct distances without collinear triples or concyclic quadruples
99	unit equilateral triangles in minimum-diameter point packings
100	diameter bounds for point sets with separated distances
101	four-point lines without five collinear points
102	many rich lines forcing a richer line
103	number of incongruent minimum-diameter point packings
104	unit circles through at least three given points
105	lines through one point set avoiding another
106	sum of side lengths in square packings
107	happy end problem
108	high-girth subgraphs of high-chromatic graphs
109	Erdős sumset conjecture
110	small finite subgraphs of uncountably chromatic graphs
111	edge deletions making finite subgraphs bipartite
112	independent sets versus transitive subtournaments
113	bipartite extremal numbers and 2-degeneracy
114	Erdős-Herzog-Piranian conjecture (polynomial level curves)
115	derivative bounds on connected polynomial lemniscates
116	area enclosed by polynomial lemniscates
117	covering groups by Abelian subgroups
118	infinite partition relations for complete graphs
119	growth of products with zeros on the unit circle
120	Erdős similarity conjecture
121	sets avoiding square products of odd lengths
122	local concentration of shifted integers
123	primitive representations using three coprime bases
124	additive representations with restricted base expansions
125	density of sums of restricted ternary and quaternary integers
126	prime divisors of products of pairwise sums
127	improvements to lower bounds for maximum cuts
128	triangles forced by dense half-size subgraphs
129	multicolor Ramsey bounds with one missing clique color
130	coloring integer-distance graphs of planar point sets
131	sets avoiding divisibility of sums of other elements
132	distances with few representations in planar point sets
133	maximum degree in triangle-free graphs of diameter 2
134	completing triangle-free graphs to diameter 2
135	distinct distances under four-point restrictions
136	edge colorings with at least five colors on every four vertices
137	powerful products of consecutive integers
138	growth of two-color van der Waerden numbers
139	Szemerédi's theorem
140	bounds for three-term-progression-free sets
141	consecutive primes in arithmetic progression
142	asymptotics for progression-free subsets of intervals
143	sparsity of separated real multiples
144	integers with two divisors within a factor of 2
145	moments of gaps between squarefree numbers
146	Erdős degeneracy conjecture
147	extremal lower bounds from bipartite minimum degree
148	counting Egyptian fraction representations of 1
149	Erdős-Nešetril conjecture (strong edge coloring)
150	number of minimal vertex cuts
151	clique transversals and triangle-free independence numbers
152	isolated sums in Sidon sumsets
153	mean square gaps in Sidon sumsets
154	distribution of large Sidon sumsets modulo integers
155	local growth of the largest Sidon set
156	small maximal Sidon sets
157	Sidon sets that are additive bases of order 3
158	density of generalized Sidon sets
159	Ramsey numbers for a 4-cycle versus a clique
160	colorings with at least three colors in every four-term progression
161	jumps in hypergraph coloring thresholds
162	logarithmic thresholds for locally balanced edge colorings
163	Burr-Erdős conjecture
164	Erdős primitive set conjecture
165	asymptotics for triangle-versus-clique Ramsey numbers
166	lower bounds for 4-clique-versus-clique Ramsey numbers
167	Tuza's conjecture
168	largest sets avoiding a number with its double and triple
169	reciprocal sums of progression-free sets
170	sparse ruler problem
171	density Hales-Jewett theorem
172	Hindman's finite sums and products conjecture
173	monochromatic triangles in two-colored planes
174	characterization of Euclidean Ramsey sets
175	Erdős squarefree conjecture
176	discrepancy bounds for finite arithmetic progressions
177	discrepancy controlled by the common difference
178	simultaneous discrepancy bounds for countably many sequences
179	many short progressions forcing a longer one
180	Erdős-Simonovits compactness conjecture
181	linear Ramsey bounds for hypercube graphs
182	edge bounds forcing regular subgraphs
183	growth of multicolor triangle Ramsey numbers
184	Erdős-Gallai cycle decomposition conjecture
185	Moser's cube problem (density-zero assertion)
186	growth of nonaveraging sets
187	long monochromatic progressions with prescribed differences
188	red unit distances versus blue unit-step progressions
189	monochromatic rectangles of every area
190	monochromatic or rainbow arithmetic progressions
191	monochromatic cliques with large logarithmic weights
192	three-term progressions in positive lattice walks
193	three collinear points in walks with finitely many steps
194	monotone progressions in arbitrary orderings of the reals
195	monotone progressions in permutations of the integers
196	monotone four-term progressions in permutations of positive integers
197	partitioning positive integers into progression-avoiding permutations
198	infinite progressions in complements of Sidon sets
199	infinite progressions outside three-term-progression-free real sets
200	length of the longest arithmetic progression of primes
201	large progression-free subsets of arbitrary integer sets
202	number of disjoint residue classes with bounded moduli
203	compositeness of all shifted multiples of powers of 2 and 3
204	nearly disjoint covering systems using divisors of one integer
205	sums of a power of 2 and an integer with few prime factors
206	eventual greediness of best Egyptian fraction underapproximations
207	Steiner triple systems of arbitrarily large girth
208	gaps between squarefree numbers
209	ordinary triangles in line arrangements
210	number of ordinary lines through planar points
211	lines determined by point sets with bounded collinearity
212	Erdős-Ulam problem (dense rational distances)
213	integral-distance point sets in general position
214	unit squares outside unit-distance-free sets
215	sets meeting the integer lattice once under every rigid motion
216	empty convex polygons in planar point sets
217	planar crescent configurations
218	comparisons and equalities of consecutive prime gaps
219	Green-Tao theorem
220	second moments of gaps between reduced residues
221	sparse additive complements of powers of 2
222	gaps between sums of two squares
223	number of diameter pairs in finite point sets
224	obtuse angles in large point sets
225	integral bounds for real-rooted trigonometric polynomials
226	entire functions preserving rationality in both directions
227	maximal terms versus maximum modulus of entire functions
228	flat Littlewood polynomials on the unit circle
229	entire functions with derivatives vanishing on prescribed discrete sets
230	maximum-modulus lower bounds for unimodular polynomials
231	Abelian squares in finite words
232	maximum density of measurable unit-distance-free sets
233	second moments of prime gaps
234	limiting distribution of normalized prime gaps
235	limiting distribution of gaps between reduced primorial residues
236	number of prime-plus-power-of-2 representations
237	unbounded representation counts using a prime and a set element
238	long sequences of consecutive primes with separated gaps
239	mean values of sign-valued multiplicative functions
240	diverging gaps in integers supported on an infinite prime set
241	largest sets with unique three-term sums
242	Erdős-Straus conjecture
243	rational reciprocal sums with near-quadratic denominator growth
244	density of primes plus geometric-sequence terms
245	sumset growth for zero-density sets
246	complete sequences generated by two coprime bases
247	transcendence of sparse binary expansions
248	uniform bounds for prime-factor counts after an integer
249	irrationality of the binary totient series
250	irrationality of the binary divisor-sum series
251	irrationality of the binary prime series
252	irrationality of factorial series with divisor-power sums
253	completeness from small successive ratios and progression intersections
254	completeness criteria using growth and fractional parts
255	unbounded discrepancy on a fixed interval
256	maximum modulus of products of binomials
257	irrationality of reciprocal Mersenne subseries
258	irrationality of divisor-count Cantor series
259	irrationality of a squarefree-weighted binary series
260	irrationality of weighted sparse binary series
261	multiple representations by weighted binary reciprocals
262	growth of universally irrational reciprocal sequences
263	irrationality under asymptotically equivalent denominator perturbations
264	irrationality under bounded denominator perturbations
265	growth of denominator sequences with two rational reciprocal sums
266	irrationality of some shifted reciprocal sum
267	irrationality of lacunary Fibonacci reciprocal sums
268	interior of the set of three shifted reciprocal sums
269	irrationality of reciprocal least-common-multiple series over smooth numbers
270	irrationality of reciprocal rising-product series
271	Stanley sequences with prescribed second term
272	set families with arithmetic-progression intersections
273	covering systems with prime-minus-one moduli
274	Herzog-Schönheim conjecture
275	covering all integers from a long covered interval
276	composite Fibonacci-type sequences without a finite prime cover
277	large divisor sums without divisor-modulus covering systems
278	extremal densities of unions of prescribed residue classes
279	covering integers by truncated prime residue classes
280	integers missed by sparse sequences of residue classes
281	uniform finite approximation to density-one residue coverings
282	termination of the odd greedy Egyptian fraction algorithm
283	prescribed polynomial sums of Egyptian fraction denominators
284	largest possible minimum Egyptian fraction denominator
285	smallest possible maximum Egyptian fraction denominator
286	Egyptian fraction representations with denominators in short intervals
287	gaps between Egyptian fraction denominators
288	integer reciprocal sums over two intervals
289	unit reciprocal sums over separated intervals
290	decreasing denominators of consecutive harmonic subsums
291	cancellation in harmonic numbers written over least common multiples
292	possible largest denominators in Egyptian fractions for 1
293	least omitted denominator in fixed-length Egyptian fractions
294	possible smallest denominators in bounded Egyptian fractions
295	number of Egyptian fraction terms with large denominators
296	disjoint Egyptian fraction representations with bounded denominators
297	counting Egyptian fraction representations with bounded denominators
298	Egyptian fractions in sets of positive density
299	bounded-gap denominator sets without a unit reciprocal subsum
300	largest bounded sets without a unit reciprocal subsum
301	sets without a reciprocal equal to a reciprocal subsum
302	sets without a reciprocal equal to two other reciprocals
303	monochromatic solutions of the three-reciprocal equation
304	length of Egyptian fraction representations of rationals
305	largest denominators needed in Egyptian fraction representations
306	Egyptian fractions with squarefree semiprime denominators
307	products of prime reciprocal sums equal to 1
308	least integer missed by bounded-denominator reciprocal sums
309	counting integer values of bounded-denominator reciprocal sums
310	reciprocal subsums with small denominators in dense sets
311	approximating 1 by bounded-denominator reciprocal sums
312	reciprocal subsums close to 1 in multisets
313	infinitude of primary pseudoperfect numbers
314	overshoot of harmonic interval sums above 1
315	extremal denominator growth in infinite Egyptian fractions
316	partitioning a reciprocal sum below 2 into sums below 1
317	small nonzero signed harmonic sums
318	zero signed reciprocal subsums in progressions and dense sets
319	large minimal zero signed reciprocal sums
320	number of distinct reciprocal subset sums
321	sets with distinct reciprocal subset sums
322	number of representations by powers of matching degree
323	density of sums of powers with few summands
324	polynomial value sets with distinct pairwise sums
325	number of integers representable as three equal-degree powers
326	minimal additive bases with quadratic growth
327	sets avoiding pairs whose sum divides their product
328	partitioning sets to reduce additive representation bounds
329	limsup growth of infinite Sidon sets
330	positive-density minimal additive bases
331	common nonzero differences of two dense square-root-growth sets
332	bounded gaps among infinitely repeated differences
333	sparse sumset covers of zero-density sets
334	representing integers as sums of two smooth numbers
335	pairs of sets with additive sumset density
336	comparison of exact and unrestricted additive basis orders
337	unbounded doubling ratios of sparse additive bases
338	restricted orders of additive bases
339	density of restricted sums from additive bases
340	growth of the Mian-Chowla sequence
341	eventual periodicity of greedy sum-free sequences
342	Ulam sequence problems
343	arithmetic progressions among subset sums of dense multisets
344	arithmetic progressions among subset sums of square-root-growth sets
345	completeness thresholds for sequences of powers
346	growth ratios of minimally robust complete sequences
347	robust density-one subset sums with doubling growth
348	completeness under prescribed numbers of deletions
349	completeness of rounded geometric sequences
350	reciprocal sums of sets with distinct subset sums
351	completeness of polynomial values perturbed by reciprocals
352	unit-area triangles in large measurable planar sets
353	unit-area configurations in infinite-measure planar sets
354	two rounded geometric sequences as weakly complete sequences
355	rational intervals among reciprocal subsums of lacunary sequences
356	many distinct consecutive sums of bounded integers
357	long sequences with all consecutive sums distinct
358	unbounded counts of consecutive-sum representations
359	greedy sequences avoiding earlier consecutive sums
360	coloring to avoid a prescribed monochromatic subset sum
361	largest sets avoiding a prescribed subset sum
362	bounds for equal subset-sum multiplicities
363	square products over disjoint intervals of length at least 4
364	three consecutive powerful numbers
365	consecutive powerful numbers and Pell equations
366	a powerful number followed by a cube-full number
367	products of powerful parts of consecutive integers
368	largest prime factors of two consecutive integers
369	long runs of smooth consecutive integers
370	consecutive integers with small largest prime factors
371	distribution of comparisons of consecutive largest prime factors
372	three decreasing consecutive largest prime factors
373	factorials that are products of smaller factorials
374	counting factorial square products by minimum length
375	Grimm's conjecture
376	central binomial coefficients coprime to 105
377	reciprocals of primes missing from central binomial coefficients
378	squarefree binomial coefficients in a row
379	large prime-power factors throughout binomial rows
380	counting integers in intervals with repeated greatest prime factors
381	growth of the count of highly composite numbers
382	lengths of intervals with repeated greatest prime factors
383	largest prime factors near prime squares
384	small prime factors of interior binomial coefficients
385	composites near an integer relative to their least prime factors
386	binomial coefficients that are products of consecutive primes
387	Erdős-Graham binomial divisor problem
388	equal products of disjoint consecutive-integer blocks
389	divisibility between adjacent equal-length integer products
390	factorial factorizations into distinct factors exceeding the argument
391	largest possible least factor in fixed-length factorial factorizations
392	fewest bounded factors in a factorial factorization
393	factorial factorizations into distinct factors in short intervals
394	average least starts of consecutive products divisible by an integer
395	reverse Littlewood-Offord problem
396	central binomial coefficients divisible by long terminal products
397	equalities between products of distinct central binomial coefficients
398	Brocard's problem
399	factorials as sums or differences of two higher powers
400	factorial divisibility and sums of factorial arguments
401	large factorial arguments dividing a prime-weighted factorial
402	Graham's GCD problem
403	powers of 2 as sums of distinct factorials
404	prime-power divisibility of sums of factorials
405	prime powers as a factorial plus a power
406	powers of 2 with only ternary digits 0 and 1
407	bounded representation counts using powers of 2 and 3
408	distribution of iterated totient stopping times
409	primes reached by iterating the totient function plus 1
410	growth of iterated divisor sums
411	eventual doubling in iterated integer-plus-totient sequences
412	intersections of iterated divisor-sum orbits
413	barriers for an integer plus its distinct-prime-factor count
414	intersections of integer-plus-divisor-count orbits
415	ordering patterns of consecutive totient values
416	asymptotic count of totient values
417	comparison of two counting functions for totient values
418	infinitely many noncototients
419	limit points of consecutive factorial divisor-count ratios
420	divisor-count ratios of nearby factorials
421	density-one sequences with distinct consecutive products
422	Hofstadter's Q-sequence
423	growth of a greedy consecutive-sum sequence
424	density of the closure under pairwise products minus 1
425	multiplicative Sidon sets and distinct higher products
426	graphs with many uniquely occurring subgraphs
427	divisible sums of consecutive primes
428	large sets of shifts simultaneously producing primes
429	infinite admissible sets of prime-producing shifts
430	composites in a greedy large-prime-factor sequence
431	additive decomposition of the primes into two infinite sets
432	density of sumsets with pairwise coprime elements
433	largest Frobenius numbers with bounded generators
434	generator sets maximizing the number of unrepresentable integers
435	Frobenius numbers of binomial-coefficient generators
436	least runs of consecutive higher-power residues
437	many square partial products of increasing integers
438	large sets whose sumset contains no square
439	monochromatic pairs summing to a square or higher power
440	growth of consecutive least common multiples
441	largest sets with bounded pairwise least common multiples
442	reciprocal least-common-multiple sums in sets with large reciprocal sums
443	common products of two integers with prescribed sums
444	large restricted divisor counts relative to reciprocal sums
445	multiplicative inverses in short intervals modulo primes
446	density of integers with divisors in a doubling interval
447	largest union-free set families
448	number of dyadic intervals occupied by divisors
449	nearby divisor pairs relative to divisor counts
450	local density of integers with divisors in a doubling interval
451	prime factors of short products of consecutive integers
452	long intervals of integers with many distinct prime factors
453	prime squares versus symmetric prime products
454	symmetric prime sums exceeding twice the central prime
455	growth of convex sequences of primes
456	least totient multiples versus least primes in progressions
457	short consecutive products divisible by all small primes
458	least-common-multiple growth between consecutive primes
459	gaps between integers supported on primes dividing the endpoints
460	reciprocal sums in a greedy pairwise-coprime construction
461	distinct smooth parts in short intervals
462	weighted sums of least prime factors in short intervals
463	composites close to an integer relative to their least prime factors
464	nondense irrational multiples of lacunary sequences
465	upper bounds for planar sets avoiding near-integer distances
466	large planar sets with all distances separated from integers
467	two prime-residue coverings of a short interval
468	new partial divisor sums and least representing integers
469	reciprocal sums of primitive pseudoperfect numbers
470	odd and primitive weird numbers
471	unbounded prime sets closed under three-prime sums
472	infinite greedy prime sequences formed from shifted sums
473	permutations with consecutive terms summing to primes
474	three-color partition relations for uncountable sets of reals
475	ordering finite-field subsets with distinct partial sums
476	Erdős-Heilbronn conjecture
477	unique additive complements of polynomial value sets
478	number of distinct factorial residues modulo a prime
479	prescribed residues of powers of 2 modulo their exponents
480	close pairs at fixed index gaps in bounded real sequences
481	repetitions in recursively generated affine sequences
482	Graham-Pollak sequences and binary digits
483	growth of Schur numbers
484	many monochromatic sums in finite colorings
485	number of nonzero terms in polynomial squares
486	logarithmic density outside truncated modular exclusions
487	least-common-multiple triples in positive-density sets
488	density ratios for sets of multiples
489	second moments of gaps between integers avoiding sparse divisors
490	product bounds for sets with distinct cross-products
491	additive functions with bounded consecutive differences
492	equidistribution within gaps of ratio-one sequences
493	representing integers as a product minus a sum
494	reconstructing finite complex sets from fixed-size subset sums
495	Littlewood conjecture
496	Diophantine approximation by an irrational quadratic form
497	Dedekind numbers (Dedekind's problem)
498	concentration of signed sums of complex numbers
499	large diagonal products in doubly stochastic matrices
500	Turán's tetrahedron problem
501	independent sets for bounded small-measure set mappings
502	maximum size of two-distance sets
503	maximum size of sets with every triangle isosceles
504	large angles forced by finite planar point sets
505	Borsuk's conjecture
506	minimum number of determined circles
507	Heilbronn triangle problem
508	Hadwiger-Nelson problem
509	covering polynomial lemniscates by circles with small total radius
510	negative values of cosine sums with distinct integer frequencies
511	large connected components of polynomial lemniscates
512	Littlewood's exponential sum problem
513	maximal-term ratios for transcendental entire functions
514	paths of rapid growth for transcendental entire functions
515	integrable reciprocal powers along paths to infinity
516	minimum and maximum modulus of lacunary entire functions
517	value distribution of lacunary entire functions
518	covering two-colored complete graphs by same-color paths
519	lower bounds for complex power sums
520	iterated-logarithm law for random multiplicative functions
521	almost-sure real-root counts for random Littlewood polynomials
522	zeros of random Littlewood polynomials in the unit disk
523	maximum modulus of random Littlewood polynomials
524	maximum on the real unit interval of random sign polynomials
525	minimum modulus of typical Littlewood polynomials
526	almost-sure coverage of the circle by random arcs
527	boundary convergence of random power series
528	connective constants for self-avoiding walks
529	displacement of random self-avoiding walks
530	large Sidon subsets of arbitrary finite real sets
531	finite bounds for monochromatic subset sums
532	Hindman's theorem
533	large triangle-free subsets of dense graphs without 5-cliques
534	largest pairwise noncoprime sets containing the upper endpoint
535	sets avoiding equal pairwise greatest common divisors
536	sets without triples having equal pairwise least common multiples
537	prime-multiple coincidences in dense sets
538	reciprocal sums under bounded prime-multiple representation counts
539	number of distinct quotients by pairwise greatest common divisors
540	Olson constant
541	residue sequences with zero sums of only one length
542	reciprocal sums of sets with large pairwise least common multiples
543	random subset sums covering finite Abelian groups
544	differences of consecutive triangle-versus-clique Ramsey numbers
545	graphs maximizing Ramsey numbers for a fixed edge count
546	Ramsey bounds in terms of edge count
547	tree Ramsey numbers bounded by twice the vertex count minus 2
548	Erdős-Sós theorem (former tree conjecture)
549	Ramsey numbers of trees with unequal bipartition classes
550	tree-versus-multipartite-graph Ramsey bounds
551	cycle-versus-clique Ramsey numbers
552	4-cycle-versus-star Ramsey numbers
553	comparison of two- and three-color triangle Ramsey numbers
554	multicolor odd-cycle versus triangle Ramsey growth
555	multicolor even-cycle Ramsey numbers
556	three-color cycle Ramsey bounds
557	multicolor tree Ramsey bounds
558	multicolor complete-bipartite Ramsey numbers
559	size Ramsey numbers of bounded-degree graphs
560	size Ramsey numbers of balanced complete bipartite graphs
561	size Ramsey numbers of star forests
562	tower growth of uniform hypergraph Ramsey numbers
563	thresholds for locally balanced edge colorings
564	double-exponential lower bounds for 3-uniform Ramsey numbers
565	exponential upper bounds for induced Ramsey numbers
566	Ramsey size linearity for locally sparse graphs
567	Ramsey size linearity for three specified graphs
568	criteria for Ramsey size linearity from trees and cliques
569	odd-cycle Ramsey bounds in terms of opposing edge count
570	sharp cycle Ramsey bounds in terms of opposing edge count
571	rational exponents conjecture
572	extremal lower bounds for even cycles
573	extremal graphs without triangles or 4-cycles
574	extremal graphs without two consecutive odd-even cycle lengths
575	extremal numbers for forbidden families containing a bipartite graph
576	extremal numbers for hypercube graphs
577	vertex-disjoint 4-cycle factors from a minimum-degree bound
578	spanning hypercubes in random graphs
579	large independent sets in dense graphs without
580	Loebl's conjecture
581	largest bipartite subgraphs of triangle-free graphs
582	4-clique-free graphs forcing monochromatic triangles
583	Gallai's path decomposition conjecture
584	dense subgraphs with pairs of edges on short common cycles
585	edge bounds forcing two cycles on the same vertices
586	covering systems with pairwise nondividing moduli
587	large sets without square subset sums
588	number of rich lines under bounded collinearity
589	large subsets without collinear triples
590	triangle partition relation at
591	triangle partition relation at
592	triangle partition relations for countable power-of-omega ordinals
593	finite subhypergraphs forced by uncountable chromatic number
594	large odd cycles in uncountably chromatic graphs
595	countable triangle-free decompositions of 4-clique-free graphs
596	finite-versus-countable edge-color Ramsey properties
597	ordinal partition relations with forbidden subgraphs
598	colorings of countable subsets realizing all colors on large sets
599	Erdős-Menger theorem (former conjecture)
600	edge thresholds forcing large books
601	infinite paths or independent sets of prescribed ordinal type
602	two-coloring almost disjoint countable families without singleton intersections
603	chromatic bounds for countable sets without two-point intersections
604	many distinct distances from one planar point
605	superlinear repeated-distance counts on the sphere
606	possible numbers of lines determined by planar points
607	possible sets of line multiplicities in planar configurations
608	edges lying in pentagons above the triangle threshold
609	short monochromatic odd cycles in multicolor complete graphs
610	upper bounds for clique transversal numbers
611	clique transversals when all maximal cliques are large
612	diameter bounds from minimum degree and forbidden cliques
613	decomposition into a bipartite graph and a bounded-degree graph
614	minimum edges forcing high degree in every small induced subgraph
615	large independent sets near the 4-clique Ramsey-Turán threshold
616	global transversals from local intersecting hypergraph conditions
617	Erdős-Gyárfás balanced coloring conjecture
618	completing low-degree triangle-free graphs to diameter 2
619	completing connected triangle-free graphs to diameter 4
620	large triangle-free induced subgraphs of 4-clique-free graphs
621	Erdős-Gallai-Tuza conjecture
622	many cycle-spanned subsets in dense regular graphs
623	infinite free sets for finite-subset mappings
624	thresholds for subset mappings realizing every value
625	chromatic versus cochromatic numbers of random graphs
626	asymptotic tradeoffs between graph girth and chromatic number
627	maximum chromatic-to-clique-number ratio
628	Erdős-Lovász Tihany conjecture
629	small bipartite graphs with large list chromatic number
630	3-choosability of planar bipartite graphs
631	5-choosability of planar graphs
632	multiples of graph list multicolorings
633	triangles allowing only square numbers of congruent pieces
634	possible numbers of congruent pieces in triangle dissections
635	large sets avoiding divisibility by long differences
636	many vertex-count and edge-count pairs in Ramsey graphs
637	induced subgraphs with many distinct degrees in Ramsey graphs
638	infinite-cardinal extensions of finite triangle-Ramsey families
639	edges outside monochromatic triangles in two-colored complete graphs
640	high-chromatic induced vertex sets of odd cycles
641	many edge-disjoint cycles on the same vertices
642	edge bounds when cycles have fewer chords than vertices
643	equal unions of disjoint pairs of uniform hyperedges
644	transversal bounds from locally two-pierceable set families
645	monochromatic three-term progressions with large common difference
646	factorials with even valuations at prescribed primes
647	barriers for an integer plus its divisor count
648	long decreasing subsequences of greatest prime factors
649	prescribed greatest prime factors of consecutive integers
650	many distinct multiples of distinct set elements in intervals
651	higher-dimensional happy end problem bounds
652	several points with few distinct distances
653	many different pinned distance counts in one point set
654	pinned Erdős distinct distances problem, no four concyclic points
655	distinct distances under centered-circle restrictions
656	shifted infinite restricted sumsets inside positive-density sets
657	distinct distances in sets without isosceles triangles
658	squares in dense subsets of the integer grid
659	few global distances under four-point local restrictions
660	distinct distances among convex polyhedron vertices
661	few cross-distances between two large planar point sets
662	triangular-lattice distance comparison (formulation unclear)
663	least prime missing from a short consecutive product
664	bounded-multiplicity transversals for nearly disjoint large sets
665	pairwise balanced designs with large blocks
666	6-cycle-free subgraphs of hypercubes
667	clique-size exponents under local edge-density constraints
668	nonuniqueness of point sets maximizing unit distances
669	extremal numbers of lines through given numbers of points
670	diameter of point sets with all distances mutually separated
671	pointwise convergence and unbounded Lebesgue functions in interpolation
672	perfect powers among products of arithmetic progressions
673	sums of ratios of consecutive divisors
674	products of self-powers equal to a self-power
675	translation recurrence in arithmetic sets
676	approximating integers by multiples of prime squares
677	distinct least common multiples of equal-length disjoint intervals
678	comparison of least common multiples of disjoint unequal intervals
679	uniform prime-factor-count bounds before an integer
680	nearby integers with large least prime factors
681	nearby composites with large least prime factors
682	rough composites inside prime gaps
683	large prime factors of binomial coefficients
684	growth of small-prime parts of binomial coefficients
685	distinct prime-factor counts of binomial coefficients
686	integers as ratios of equal-length consecutive products
687	Jacobsthal function and prime-residue interval coverings
688	covering intervals using only large prime moduli
689	double coverings of intervals by prime residue classes
690	unimodality of the distribution of the th smallest prime factor
691	criteria for sets of multiples to have density 1
692	unimodality of the density of exactly one interval divisor
693	gaps between integers with a divisor in a doubling interval
694	ratios of greatest to least totient preimages
695	growth of prime chains with successive divisibility
696	lengths of divisor chains with successive congruences
697	density thresholds for divisors in a prescribed residue class
698	growing common divisors of pairs of binomial coefficients
699	large common prime divisors of pairs of binomial coefficients
700	common divisors of an integer and its binomial coefficients
701	Chvátal's intersecting-family conjecture
702	singleton intersections in large uniform set families
703	set families avoiding a prescribed intersection size
704	growth of higher-dimensional unit-distance chromatic numbers
705	3-colorability of high-girth unit-distance graphs in the plane
706	chromatic bounds for graphs defined by finitely many distances
707	extending Sidon sets to perfect difference sets
708	covering prescribed products by products from intervals
709	distinct representatives divisible by prescribed integers
710	short intervals containing distinct multiples of every small integer
711	uniform interval bounds for distinct divisible representatives
712	Turán densities of complete uniform hypergraphs
713	power-law asymptotics and rational exponents for bipartite extremal numbers
714	extremal lower bounds for balanced complete bipartite graphs
715	cubic subgraphs of regular graphs
716	Ruzsa-Szemerédi theorem ((6,3)-theorem)
717	chromatic number versus largest topological clique
718	quadratic degree bounds forcing complete graph subdivisions
719	decomposing uniform hypergraphs into edges and small cliques
720	size Ramsey numbers of paths and cycles
721	off-diagonal van der Waerden numbers
722	existence of Steiner systems under divisibility conditions
723	prime power conjecture for finite projective planes
724	many mutually orthogonal Latin squares
725	asymptotic enumeration of Latin rectangles
726	reciprocal sums of primes with residues in an upper half-interval
727	factorial-square divisibility with shifted arguments
728	factorial divisibility with sums of arguments exceeding the total
729	factorial ratios with denominators supported on bounded primes
730	central binomial coefficients with identical prime supports
731	least nondivisor of a central binomial coefficient
732	possible block-size sequences of pairwise balanced designs
733	possible line-multiplicity sequences of planar point sets
734	pairwise balanced designs with bounded block-size multiplicities
735	magic configurations with positive point weights and equal line sums
736	arbitrarily high chromatic graphs with prescribed finite subgraphs
737	a fixed edge on all sufficiently long cycles
738	induced trees in infinitely chromatic triangle-free graphs
739	subgraphs with prescribed smaller infinite chromatic numbers
740	preserving infinite chromatic number while eliminating short odd cycles
741	partitioning sets while preserving positive sumset density
742	Murty-Simon conjecture
743	tree packing conjecture (Gyárfás-Lehel)
744	edge deletions making chromatic-critical graphs bipartite
745	second-largest components in critical random graphs
746	Hamiltonian cycles in random graphs above the connectivity threshold
747	perfect matching thresholds in random 3-uniform hypergraphs
748	Cameron-Erdős conjecture
749	almost-complete sumsets with bounded representation counts
750	infinitely chromatic graphs with nearly half-sized independent sets
751	widely separated cycle lengths in 4-chromatic graphs
752	many cycle lengths from minimum degree and girth
753	list chromatic numbers of a graph and its complement
754	many equal distances from each point in four dimensions
755	unit equilateral triangle counts in six dimensions
756	many distances each occurring more often than the number of points
757	large Sidon subsets under local difference-set constraints
758	maximum cochromatic numbers of small graphs
759	cochromatic numbers on surfaces of prescribed genus
760	large cochromatic subgraphs of high-chromatic graphs
761	dichromatic numbers versus chromatic and cochromatic numbers
762	chromatic versus cochromatic numbers of 5-clique-free graphs
763	Erdős-Fuchs theorem (bounded-error case)
764	bounded-error linear growth of cumulative triple-sum counts
765	asymptotic extremal number of the 4-cycle
766	minimum extremal numbers for fixed vertex and edge counts
767	edge bounds forcing cycles with many chords at one vertex
768	density of integers with divisors congruent to 1 modulo their prime factors
769	tiling cubes by prescribed numbers of smaller cubes
770	common divisors of initial power-minus-one values
771	large subsets avoiding any one prescribed sum
772	large Sidon subsets of bounded-representation sets
773	largest Sidon subsets of the squares
774	partitioning proportionately dissociated sets into dissociated sets
775	many maximal clique sizes in 3-uniform hypergraphs
776	antichains with many sizes and prescribed size multiplicities
777	edge counts of comparability graphs of set families
778	clique-size and degree games on complete graph edges
779	prime sums involving a primorial
780	multicolor Ramsey numbers for uniform hypergraph matchings
781	monochromatic descending waves
782	quasi-progressions and additive cubes among squares
783	minimizing uncovered integers for pairwise coprime divisor sets
784	integers avoiding divisor sets with bounded reciprocal sums
785	error terms for exact additive complements
786	dense sets with uniquely determined product lengths
787	large subsets with pairwise sums outside the original set
788	subsets whose pairwise sums avoid a prescribed higher interval set
789	maximum size of an admissible set
790	large subsets avoiding equations between an element and distinct-element sums
791	minimum size of additive two-bases for finite intervals
792	large sum-free subsets of arbitrary integer sets
793	sets avoiding divisibility by products of other elements
794	small dense subhypergraphs above a tripartite edge threshold
795	largest sets with distinct subset products
796	large sets with bounded multiplicative representation counts
797	acyclic chromatic number in terms of maximum degree
798	covering integer grids by lines through selected grid points
799	list chromatic number of typical graphs
800	linear Ramsey bounds when high-degree vertices are nonadjacent
801	small dense vertex sets forced by small independence number
802	independent sets in clique-free graphs of prescribed average degree
803	dense almost-regular subgraphs of prescribed size
804	global independence bounds from local independent sets
805	graphs with cliques and independent sets in every small vertex set
806	small sumset covers of square-root-sized sets
807	biclique partition number of random graphs
808	sum-product bounds restricted to graph edges
809	Burr-Erdős-Graham-Sós conjecture
810	dense edge-colored graphs with every 4-cycle rainbow
811	rainbow subgraphs in balanced complete graph colorings
812	ratio and difference growth of consecutive diagonal Ramsey numbers
813	large cliques when every seven vertices contain a triangle
814	small high-minimum-degree induced subgraphs above degeneracy thresholds
815	prescribed cycles in minimally non-2-degenerate graphs
816	equal-degree endpoints of three-edge paths in dense graphs
817	sets whose subset sums avoid arithmetic progressions
818	large product sets from small sumsets
819	largest truncated sumsets of square-root-sized sets
820	coprime pairs of power-minus-one values
821	large multiplicities of totient values
822	density of the integer-plus-totient image
823	ratios of integers with equal divisor sums
824	counting coprime pairs with equal divisor sums
825	sufficient abundance for pseudoperfectness
826	uniform divisor-count bounds after an integer
827	large subsets with all circumradii distinct
828	totient divisibility of shifted arguments
829	bounds for the number of two-cube representations
830	infinitely many amicable pairs
831	distinct circumradii determined by planar point triples
832	minimum edge count for uniform hypergraphs of given chromatic number
833	high-degree vertices in non-2-colorable uniform hypergraphs
834	3-critical triple systems with large minimum degree
835	subset colorings with every local palette complete in Johnson graphs
836	intersecting non-2-colorable hypergraphs with large edge intersections
837	jumping densities of 3-uniform hypergraphs
838	number of convex subsets of planar point sets
839	sparsity of sequences avoiding earlier consecutive sums
840	largest quasi-Sidon subsets of intervals
841	completing integers to squares using products of nearby integers
842	3-coloring a Hamiltonian cycle with disjoint triangles
843	Ramsey 2-completeness of the squares
844	largest sets with no squarefree pairwise products
845	density of sums of comparable integers supported on 2 and 3
846	finite partitions into sets without collinear triples
847	finite partitions into three-term-progression-free sets
848	largest sets whose pairwise products plus 1 are not squarefree
849	prescribed multiplicities among binomial coefficients
850	Erdős-Woods conjecture (three-term version)
851	almost-all representations by a power of 2 and an almost prime
852	long runs of distinct prime gaps
853	least missing even prime gap
854	missing and distinct gaps between reduced primorial residues
855	second Hardy-Littlewood conjecture
856	reciprocal sums without equal pairwise least common multiples
857	Erdős-Szemerédi sunflower conjecture
858	reciprocal sums of sets avoiding rough integer ratios
859	density of integers representing a target by distinct divisors
860	short intervals with distinct representatives divisible by small primes
861	enumeration of Sidon sets relative to their maximum size
862	enumeration of maximal Sidon sets
863	comparison of bounded sum-representation and difference-representation sets
864	large sets with only one repeated pair sum
865	three elements and all their pair sums inside a dense set
866	pair-sum cliques inside dense integer sets
867	large sets avoiding membership of consecutive sums
868	minimal additive subbases from diverging representation counts
869	minimal subbases in unions of disjoint additive bases
870	minimal higher-order subbases under logarithmic representation bounds
871	partitioning rich additive bases into two bases
872	length of the primitive-set selection game
873	counts of bounded least common multiples of consecutive set elements
874	largest sets with disjoint subset-sum lengths
875	growth and gaps of infinite sets with disjoint subset-sum lengths
876	gaps in infinite sets avoiding distinct-element sum equations
877	enumeration of maximal sum-free sets
878	extremal sums of coprime integers with prescribed prime support
879	maximum sums of pairwise coprime bounded integers
880	bounded gaps among restricted sums from additive bases
881	deleting infinite sets from minimal higher-order bases
882	sets whose nonempty subset sums form a primitive set
883	odd cycles and tripartite subgraphs in coprimality graphs
884	reciprocal divisor differences versus consecutive divisor differences
885	common differences between factor pairs of distinct integers
886	few divisors just above the square root
887	uniform divisor bounds in short intervals above the square root
888	sets where square quadruple products force equal cross-products
889	unbounded new prime factors along consecutive integers
890	prime-factor counts in blocks of consecutive integers
891	integers with many prime factors in primorial-length intervals
892	growth and density criteria for primitive sequences
893	growth of cumulative Mersenne divisor counts
894	colorings avoiding lacunary differences
895	independent Schur triples in triangle-free graphs
896	products with unique cross-factor representations
897	large consecutive jumps of additive functions
898	Erdős-Mordell theorem
899	difference-set growth of zero-density sets
900	long paths in supercritical random graphs
901	minimum edges in non-2-colorable uniform hypergraphs
902	minimum tournament size with all small sets externally dominated
903	gaps in block counts above projective plane designs
904	large degree sums on cliques at the Turán threshold
905	large books above the triangle extremal threshold
906	dense zero sets along every infinite subsequence of derivatives
907	decomposing functions with continuous differences
908	decomposing functions with measurable differences
909	spaces whose squares have the same positive finite dimension
910	proper nonhomeomorphic connected subsets of Euclidean connected sets
911	superlinear size Ramsey bounds from high average degree
912	distinct exponents in prime factorizations of factorials
913	consecutive-integer products with distinct prime exponents
914	Hajnal-Szemerédi theorem
915	many disjoint paths between two vertices above an edge threshold
916	cycles with an external vertex adjacent to three cycle vertices
917	maximum edge counts in chromatic-critical graphs
918	infinite chromatic number with smaller subgraphs countably colorable
919	chromatic number versus ordinal type in infinite graphs
920	high-chromatic graphs avoiding a fixed clique
921	largest odd girth for fixed chromatic number
922	chromatic bounds from nearly half-sized independent sets in subgraphs
923	high-chromatic triangle-free subgraphs
924	clique-free graphs forcing monochromatic cliques
925	large independent sets in graphs not Ramsey for triangles
926	extremal numbers for augmented subdivided cliques
927	number of distinct maximal clique sizes
928	joint distribution of consecutive greatest prime factors
929	small primes covering blocks of consecutive integers
930	perfect powers among products over disjoint long intervals
931	disjoint consecutive products with the same prime factors
932	multiple smooth integers inside prime gaps
933	large 2-and-3-smooth parts of consecutive-integer products
934	distant edges forced by edge count and maximum degree
935	powerful parts of products of consecutive integers
936	powerful neighbors of powers of 2 and factorials
937	four-term progressions of coprime powerful numbers
938	three-term progressions of consecutive powerful numbers
939	additive relations between coprime higher-powerful numbers
940	sums of higher-powerful numbers, including cube-full numbers
941	representing large integers as three powerful numbers
942	powerful numbers between consecutive squares
943	number of representations as two powerful numbers
944	vertex-critical graphs with no small critical edge sets
945	long runs of distinct divisor counts
946	Erdős-Mirsky conjecture
947	Mirsky-Newman theorem
948	growth bounds for subset sums missing a color
949	large pair-sum sets outside real sum-free sets
950	reciprocal distances to preceding primes
951	counting multiplicatively independent generators of separated products
952	Gaussian moat problem
953	measure of planar sets without integer distances
954	error bounds for a greedy cumulative pair-sum construction
955	Erdős-Granville-Pomerance-Spiro conjecture
956	unit separations between disjoint translates of a convex set
957	product of shortest- and longest-distance multiplicities
958	characterizing point sets with consecutive distance multiplicities
959	gap between the two largest distance multiplicities
960	cliques of ordinary lines under bounded collinearity
961	maximum runs of smooth consecutive integers beyond the smoothness bound
962	long consecutive blocks each with a prime factor exceeding the length
963	large subsets with distinct subset sums in arbitrary real sets
964	density of ratios of consecutive divisor counts
965	uncountable sets with monochromatic restricted pair sums
966	Ramsey progression-free sets forcing shorter monochromatic progressions
967	zero-free Dirichlet series with positive reciprocal coefficients
968	density of increases in the nth prime divided by its index
969	error term in counting squarefree integers
970	growth of the Jacobsthal function by number of prime factors
971	large least primes in many residue classes
972	prime values of irrational multiples of primes rounded down
973	exponentially small consecutive complex power sums
974	repeated zero blocks in power sums forcing roots of unity
975	average divisor counts of irreducible polynomial values
976	greatest prime factors of products of polynomial values
977	growth of greatest prime factors of Mersenne numbers
978	power-free values of irreducible polynomials
979	unbounded representation counts by matching-degree prime powers
980	average least higher-power nonresidues over primes
981	average thresholds for cancellation in quadratic character sums
982	many distinct distances from a convex polygon vertex
983	small prime supports containing many set elements
984	colorings with short monochromatic progressions relative to their starts
985	prime primitive roots below every prime modulus
986	lower bounds for off-diagonal clique Ramsey numbers
987	large Fourier sums at some integer frequency
988	unbounded spherical cap discrepancy
989	disk-counting discrepancy of infinite planar sequences
990	angular distribution of polynomial roots in terms of sparsity
991	equidistribution of sphere points maximizing products of distances
992	metric discrepancy bounds for integer sequences
993	unimodality of independent-set sequences of forests
994	metric equidistribution in measurable sets for irrational rotations
995	metric sums of square-integrable functions along lacunary sequences
996	Fourier approximation criteria for lacunary equidistribution
997	failure of well-distribution for prime multiples
998	bounded remainder intervals for irrational rotations
999	Duffin-Schaeffer conjecture
1000	vanishing average proportions of new reduced denominators
1001	limiting measures of rational approximation over denominator intervals
1002	limiting distribution of logarithmically normalized fractional-part sums
1003	equal totients of consecutive integers
1004	long runs of distinct totient values
1005	local ordering of Farey numerators and denominators
1006	acyclic orientations remaining acyclic after any single reversal
1007	fewest edges in a graph of Euclidean dimension 4
1008	large 4-cycle-free subgraphs of arbitrary graphs
1009	packing edge-disjoint triangles above the extremal threshold
1010	triangle counts just above the extremal threshold
1011	edge thresholds forcing triangles at prescribed chromatic number
1012	long cycles forced by edge counts
1013	minimum order of triangle-free graphs with prescribed chromatic number
1014	ratios of consecutive off-diagonal Ramsey numbers
1015	covering vertices by disjoint monochromatic cliques
1016	minimum edge counts of pancyclic graphs
1017	clique partition numbers of dense graphs
1018	bounded-size nonplanar subgraphs of dense graphs
1019	large maximal planar subgraphs above an edge threshold
1020	Erdős matching conjecture
1021	extremal numbers for subdivided complete graphs
1022	property B from local sparsity of large set families
1023	families with no member a union of other members
1024	independent sets in linear triple systems
1025	large free sets for pair-to-element mappings
1026	large sums on monotone subsequences
1027	many two-colorings avoiding monochromatic edges in uniform hypergraphs
1028	discrepancy of signed pairs on induced vertex sets
1029	improving the probabilistic lower bound for diagonal Ramsey numbers
1030	ratios of nearly diagonal Ramsey numbers
1031	nontrivial regular induced subgraphs of Ramsey graphs
1032	large minimum degree in edge-critical 4-chromatic graphs
1033	large triangle degree sums above the extremal threshold
1034	triangles with many vertices adjacent to two of their vertices
1035	spanning hypercubes from large minimum degree
1036	exponentially many nonisomorphic induced subgraphs of Ramsey graphs
1037	large homogeneous sets from nearly distinct vertex degrees
1038	length of real polynomial sublevel sets with bounded real roots
1039	large disks inside polynomial lemniscates
1040	area of polynomial lemniscates versus transfinite diameter
1041	short root-connecting paths inside polynomial lemniscates
1042	components of polynomial lemniscates versus transfinite diameter
1043	short projections of polynomial lemniscates
1044	longest component boundaries of polynomial lemniscates
1045	maximum product of distances under a diameter bound
1046	enclosing disks for connected polynomial lemniscates
1047	convexity of small polynomial sublevel components
1048	large lemniscate components from bounded roots
1049	irrationality of rational-parameter Lambert series
1050	irrationality of the reciprocal power-of-2-minus-3 series
1051	irrationality of reciprocal products with doubly exponential growth
1052	finiteness of unitary perfect numbers
1053	abundancy ratios of multiply perfect numbers
1054	least integers with prescribed sums of their smallest divisors
1055	recursive classes defined by prime factors of a prime plus 1
1056	consecutive interval products congruent to 1 modulo a prime
1057	growth of the count of Carmichael numbers
1058	factorials plus 1 supported on two nearby primes
1059	primes remaining composite after subtracting any smaller factorial
1060	multiplicities of the integer-times-divisor-sum map
1061	additive coincidences of the divisor function
1062	largest sets without an element dividing two others
1063	binomial coefficients divisible by almost all numerator-block terms
1064	totient comparisons after subtracting the totient
1065	primes with prime odd parts after subtracting 1
1066	large independent sets in planar contact graphs
1067	infinitely connected uncountably chromatic subgraphs
1068	countable infinitely connected subgraphs of uncountably chromatic graphs
1069	Szemerédi-Trotter theorem
1070	large unit-distance-free subsets of arbitrary planar point sets
1071	maximal disjoint families of unit segments
1072	least factorials congruent to minus 1 modulo primes
1073	composite divisors of factorials plus 1
1074	residue restrictions on prime divisors of factorials plus 1
1075	jumping hypergraph densities above the multipartite threshold
1076	extremal triple systems avoiding vertices with edges
1077	large dense almost-regular subgraphs
1078	cliques forced by high multipartite minimum degree
1079	dense neighborhoods in graphs at clique extremal thresholds
1080	6-cycles in unbalanced bipartite graphs
1081	density of sums of two powerful numbers
1082	distinct and pinned distances without collinear triples
1083	minimum distinct distances in higher dimensions
1084	contact number problem
1085	unit-distance counts in higher dimensions
1086	maximum number of equal-area triangles
1087	quadruples containing repeated distances
1088	large subsets with all pairwise distances distinct
1089	point-set sizes forcing prescribed numbers of distances
1090	finite point sets forcing rich monochromatic lines
1091	chords in odd cycles of 4-chromatic graphs
1092	global chromatic bounds from nearly colorable finite subgraphs
1093	deficiency of binomial coefficients without small prime factors
1094	least prime factors of binomial coefficients
1095	least binomial coefficients with no small prime factors
1096	gaps between sums of powers of a real base near 1
1097	distinct common differences of three-term progressions
1098	clique bounds for noncommuting graphs of groups
1099	sums of powers of relative gaps between consecutive divisors
1100	coprime consecutive divisor pairs
1101	gaps in integers avoiding sparse pairwise coprime divisors
1102	growth of shift sets with squarefree exclusion or recurrence
1103	growth of infinite sets with squarefree sumsets
1104	maximum chromatic number of triangle-free graphs
1105	anti-Ramsey numbers of paths and cycles
1106	distinct prime factors of partition-function values
1107	representing integers by boundedly many higher-powerful numbers
1108	perfect powers and powerful numbers among distinct factorial sums
1109	largest finite sets with squarefree sumsets
1110	primitive additive representations using two coprime bases
1111	anticomplete high-chromatic sets in graphs of bounded clique number
1112	bounded-gap sequences whose iterated sumsets avoid lacunary sets
1113	Sierpiński numbers without finite prime covering sets
1114	spacing of critical points for equally spaced real roots
1115	short paths of growth to infinity for entire functions
1116	incomparable counting functions for distinct meromorphic values
1117	number of maximum-modulus points on circles
1118	growth of entire functions with finite-area superlevel sets
1119	cardinality of families of entire functions with few pointwise values
1120	short paths from the origin to the unit circle in polynomial lemniscates
1121	enclosing circles for nonseparable circle families
1122	additive functions decreasing on a zero-density set
1123	nonisomorphism of Boolean algebras modulo density ideals
1124	Tarski's circle-squaring problem
1125	monotonicity from a three-point functional inequality
1126	almost-everywhere solutions of Cauchy's functional equation
1127	countable partitions of Euclidean space into distinct-distance sets
1128	countably infinite monochromatic boxes in uncountable products
1129	interpolation nodes minimizing the Lebesgue constant
1130	local maxima of interpolation Lebesgue functions
1131	minimum integrated squares of Lagrange basis polynomials
1132	pointwise logarithmic growth of interpolation Lebesgue functions
1133	large norms forced by nearly full interpolation
1134	density of the closure under three affine maps
1135	Collatz conjecture
1136	dense sets with no pair summing to a power of 2
1137	adjacent prime-gap products versus maximum prime gaps
1138	prime counts on the scale of maximum prime gaps
1139	large gaps between integers with at most two prime factors
1140	integers remaining prime after subtracting twice any smaller square
1141	integers remaining prime after coprime square subtractions
1142	integers remaining prime after every smaller power-of-2 subtraction
1143	minimum prime-multiple counts in short intervals
1144	large positive fluctuations of random completely multiplicative sums
1145	representation counts for asymptotically equal additive complements
1146	essential-component property of integers supported on 2 and 3
1147	additive bases defined by quadratic Diophantine approximation
1148	bounded representations as two squares minus a square
1149	coprimality of integers with their rounded noninteger powers
1150	uniform maximum-modulus lower bounds for Littlewood polynomials
1151	prescribed limit sets for interpolation at Chebyshev nodes
1152	almost-everywhere divergence of near-degree interpolation
1153	local lower bounds for interpolation Lebesgue functions
1154	Hausdorff dimensions of subrings and subfields of the reals
1155	random triangle removal process
1156	concentration of chromatic numbers of random graphs
1157	extremal hypergraphs avoiding prescribed vertex and edge counts
1158	extremal lower bounds for complete multipartite uniform hypergraphs
1159	bounded-intersection blocking sets in projective planes
1160	maximal counts of groups of bounded order
1161	most frequent element orders in symmetric groups
1162	enumeration and order distribution of symmetric-group subgroups
1163	arithmetic structure of symmetric-group subgroup orders
1164	fully covered disks of 2-dimensional random walks
1165	number of favorite sites of 2-dimensional random walks
1166	number of sites ever favorite in 2-dimensional random walks
1167	negative stepping-up problem
1168	negative countable-color partition relations without continuum assumptions
1169	triangle partition relations for the square of the first uncountable ordinal
1170	consistency of strong partition relations below the second uncountable ordinal
1171	finite-color partition relations at the square of the first uncountable ordinal
1172	ordinal partition relations under the generalized continuum hypothesis
1173	large free sets for almost disjoint set mappings
1174	clique-free graphs forcing countably colored monochromatic cliques
1175	uncountably chromatic triangle-free subgraphs
1176	edge colorings realizing every edge color within some vertex color
1177	forbidden finite subhypergraphs at uncountable chromatic numbers
1178	Brown-Erdős-Sós conjecture for uniform hypergraphs
1179	uniform representation counts for random subset sums in finite groups
1180	bounded sums of inverses of small residues modulo primes
1181	least primes missing from logarithmic-length consecutive products
1182	triangle-good connected graphs and their edge counts
1183	large monochromatic families with union and intersection closure
1184	large-prime-factor counts in polynomial-scale intervals
1185	long progressions with common differences from a prescribed difference set
1186	density of monochromatic arithmetic progressions
1187	monochromatic prime progressions and prime common differences
1188	enumeration of minimal distinct covering systems
1189	irreducible covering modulus sets
1190	reciprocal sums of large disjoint congruence moduli
1191	logarithmic improvements in infinite Sidon set density
1192	additive bases with small mean-square representation counts
1193	density of equality to a monotone representation-count function
1194	growth in perfect difference bases
1195	measure growth of real sets avoiding integer ratios
1196	weighted reciprocal sums of primitive sets with large minimum element
1197	integer dilates of positive-measure sets meeting scaled integers
1198	monochromatic sums of disjoint products from an infinite set
1199	infinite sets with monochromatic sumsets
1200	prime-residue interval coverings with bounded reciprocal modulus sum
1201	large prime factors in almost all long consecutive products
1202	integers avoiding half the residues for each prescribed prime
1203	large normalized prime-factor counts after an integer
1204	minimum diameter and mean of admissible prime-tuples
1205	maximal uniform multiplicity of residue coverings
1206	large Sidon subsets of the cubes
1207	large subsets without isosceles triangles
1208	large subsets with all distances distinct
1209	prime or squarefree translates of fast-growing sequences
1210	reciprocal distances from an endpoint to pairwise coprime integers
1211	logarithmic density of monochromatic subset sums
1212	infinite paths through visible lattice points avoiding prime pairs
1213	repeated consecutive sums in long bounded-gap sequences
1214	identical prime supports of all power-minus-one sequences
1215	short paths in sublevel sets for polynomials with roots on the unit circle
1216	large transitive subtournaments
1217	long divisibility chains in sets of positive logarithmic density
1218	negative triple partition relations under the generalized continuum hypothesis
1219	partition relations for sums of increasing cardinal exponentials
1220	partition relations at -inaccessible singular cardinals
1221	extremal spacing ratios of circular sequences

Recent results on individual problems include the following.

For problem 30, Carter et al. (2026) improved the error-term coefficient in the upper bound for the maximum size of a Sidon set in . This does not settle whether every positive power is an admissible error bound.

Problem 59 concerns enumeration of graphs with a forbidden subgraph. Yang (2026) established the case of labeled square-free graphs. This does not resolve the general formulation.

Problem 138 concerns the growth of two-color van der Waerden numbers. The three-color numerical improvements of Kim (2026) concern a different parameter choice and do not settle that problem.

Problem 302 asks how large a subset can be if no distinct satisfy . Writing for the maximum cardinality, the problem asks in particular whether , with as in little-O notation (Erdős and Graham 1980, Bloom 2026). A Lean development directed by Sodelin (2026), with the argument, code, and formalization produced by ChatGPT/Codex, reported , , and . The first relation follows from an isolated component in the hypergraph of forbidden triples containing the five vertices 122, 183, 244, 366, and 732. Every admissible selection from this component contains at most three of them and can be replaced by 122, 183, and 244. Exact certificates show that neither 733 nor 734 belongs to a forbidden triple on . The values of for , 2, ... begin 1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 10, 11, ... (OEIS A390395). This tabulation reaches , so the relations above give , , and . The development proves the three relations but not the external value . As of Sep. 25, 2026, independent human review had not established either that the formal statement matches the classical problem or that the finite result is new. The asymptotic Problem 302 remained open (VibeMathed 2026c).

Problem 433 concerns the largest Frobenius number for a fixed number of generators bounded by a parameter . Its asymptotic value was proved by Dixmier (1990). Chen et al. (2026) obtained a sharp finite upper bound for individual generating sets.

Problem 793 asks for the maximum cardinality of a subset of such that whenever , , and are distinct members of . The recorded question is whether for some constant . Bloom (2026) reports that GPT-5.6 Sol, prompted by Przemek Chojecki, gave a Lean-verified argument with

Here is the prime counting function, and denotes a quantity tending to 0 as in little-O notation. This determines the proposed constant as (Bloom 2026, VibeMathed 2026b). No independent specialist review had been reported as of Sep. 9, 2026.

Problem 940 asks, for every integer , whether infinitely many positive integers are not sums of at most -full, or -powerful, numbers, meaning numbers such that implies for every prime number . It also asks whether the integers representable by at most such summands have natural density zero (Bloom 2026). For , the summands are cube-full numbers. Beyer de Ryke (2026) proved the stronger conclusion that the positive integers not representable by at most three cube-full numbers have positive lower natural density. This resolves the first question for , while the density-zero question for remains open.

Problem 1154 concerns the possible Hausdorff dimensions of subfields of . Assuming the axiom of constructibility for the constructible universe, Wang (2026) constructed proper subfields of that are coanalytic sets with every Hausdorff dimension in . The unrestricted existence question in ZFC remains open.

For an integer , let be the least integer such that the product has no prime factor in the interval . Erdős (1979) conjectured that grows faster than every fixed power of . Van Doorn and Tang (2026) reported the stronger bound

for all sufficiently large . The paper credits ChatGPT 5.5 Pro with the core idea of adapting Konyagin's argument and supplies supporting Lean formalizations produced by Aristotle. Independent specialist review and an external audit of the formalization had not been reported as of Sep. 21, 2026 (VibeMathed 2026a).

Problem 682 concerns gaps between consecutive primes in which every intervening integer has a least prime factor smaller than the gap. Mossinghoff and Trudgian (2026) obtained a conditional counting asymptotic for these gaps, assuming the Dickson-Hardy-Littlewood prime-tuples conjecture.

Problem 789 asks for large admissible sets, whose subset sums determine the number of summands. The Erdős-Nicolas-Sárközy conjecture instead counts all such subsets of .