The Erdős problems are a large collection of questions posed or promoted by Paul Erdős and his collaborators (Bloom 2026d). They range across number theory, combinatorics, graph theory, geometry, and related areas. Bloom's online Erdős Problems database listed 1217 numbered problems as of Sep. 9, 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 links selected problem numbers to their corresponding named problems or related mathematical topics (Bloom 2026d).
)Recent results on individual problems include the following.
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 2026a). 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 374 asks for the orders of growth of the counting functions for factorial square products requiring at least three and at most six factors (Bloom 2026f). Yudin (2026) determined these orders of growth and confirmed the conjectured linear growth for integers requiring six factors. The existence of limiting densities for integers requiring four, five, or six factors remains open.
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 (2026b) 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 2026b, VibeMathed 2026b). No independent specialist review had been reported
as of Sep. 9, 2026.
Problem 809, the Burr-Erdős-Graham-Sós conjecture, was resolved by Bucić et al. (2026) for odd cycle
graphs of length at least 9 and by Shahab (2026) for the remaining case of the
cycle graph 
(Bloom 2026c).
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 2026e). 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.
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).
