The Schur number
is the largest integer for which the integers 1, 2, ...,
can be partitioned into sum-free sets, where is a positive
integer and equal summands are allowed (Fredricksen and Sweet 2000). is guaranteed to exist for each by Schur's problem. Note
the definition of the Schur number as the smallest integer for which such a set
partitiondoes not exist is also prevalent in the literature (OEIS A030126; Fredricksen and Sweet 2000).
which is sharp for ,
2, and 3 (Guy 1994). The Schur numbers also satisfy the inequality
(2)
for and some positive constant (Abbott and Moser 1966, Abbott and Hanson
1972, Exoo 1994). Schur's Ramsey theorem
also shows that
(3)
where
is the -color
Ramsey number for a triangle
graph. The known Schur numbers are 1, 4, 13, 44, 160 (OEIS A045652).
The value of
is due to Baumert (Golomb and Baumert 1965, Abbott and Hanson 1972). Exoo (1994)
established the lower bound, and Heule (2018) proved that . Fredricksen and Sweet (2000) established the lower
bounds
and . Rowley (2021) improved
the latter to .
Chen et al. (2026) improved the sixth lower bound to by giving an explicit partition
of into six sum-free
sets. Their construction also gives , , , and the Ramsey
number bound .
A generalized Schur number instead prescribes a linear equation such as . It uses the least unavoidable endpoint, rather than
the largest avoidable endpoint used for (Khamseh 2020).
Abbott, H. L. and Hanson, D. "A Problem of Schur and Its Generalizations." Acta Arith.20, 175-187, 1972. https://doi.org/10.4064/aa-20-2-175-187.Abbott,
H. L. and Moser, L. "Sum-Free Sets of Integers." Acta Arith.11,
393-396, 1966. https://doi.org/10.4064/aa-11-4-393-396.Beutelspacher,
A. and Brestovansky, W. "Generalized Schur Numbers." In Combinatorial
Theory. Proceedings of a Conference Held at Schloss Rauischholzhausen, May 6-9, 1982.
Berlin, Germany: Springer-Verlag, pp. 30-38, 1982.Chen, Z.; Zhu,
L.; and Long, F. "A New Lower Bound for the Sixth Schur Number: ." 22 Oct 2026. https://arxiv.org/abs/2610.31362.Exoo,
G. "A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers of ." Elec. J. Combin.1, R8, 1994. https://doi.org/10.37236/1188.Fredricksen,
H. "Schur Numbers and the Ramsey Numbers ." J. Combin. Theory Ser. A27,
376-377, 1979.Fredricksen, H. and Sweet, M. M. "Symmetric
Sum-Free Partitions and Lower Bounds for Schur Numbers." Elec. J. Combin.7,
R32, 2000. https://doi.org/10.37236/1510.Golomb,
S. W. and Baumert, L. D. "Backtrack Programming." J. Ass.
Comp. Machinery12, 516-524, 1965. https://doi.org/10.1145/321296.321300.Guy,
R. K. "Schur's Problem. Partitioning Integers into Sum-Free Classes"
and "The Modular Version of Schur's Problem." §E11 and E12 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 209-212,
1994.Heule, M. J. H. "Schur Number Five." In Proc.
Thirty-Second AAAI Conference on Artificial Intelligence (Ed. S. A. McIlraith
and K. Q. Weinberger). Palo Alto, CA: AAAI Press, pp. 6598-6606, 2018.
https://doi.org/10.1609/aaai.v32i1.12209.Khamseh,
A. "On Generalized Schur Numbers of the Equation ." J. Math.2020, 7069730, 2020. https://doi.org/10.1155/2020/7069730.Numberphile.
"Schur Numbers (the World's Biggest Proof)." Nov. 3, 2025. https://www.youtube.com/watch?v=57V8Ud7PL8k.Radziszowski,
S. P. "Small Ramsey Numbers." Elec. J. Combin., Dynamic Survey
DS1, Apr. 24, 2026. https://doi.org/10.37236/21.Rowley,
F. "An Improved Lower Bound for and Some Interesting Templates." 8 Jul 2021. https://arxiv.org/abs/2107.03560.Schur,
I. "Über die Kongruenz (mod )." Jahresber. Deutsche Math.-Verein.25,
114-116, 1916.Sloane, N. J. A. Sequences A030126
and A045652 in "The On-Line Encyclopedia
of Integer Sequences."Whitehead, E. G. "The Ramsey Number
." Disc. Math.4,
389-396, 1973.
is the largest integer 
for which the integers 1, 2, ...,
can be partitioned into 
sum-free sets, where 
is a positive
integer and equal summands are allowed (Fredricksen and Sweet 2000). 
is guaranteed to exist for each 
by Schur's problem. Note
the definition of the Schur number as the smallest integer 
for which such a set
partition does not exist is also prevalent in the literature (OEIS A030126; Fredricksen and Sweet 2000).
,
2, and 3 (Guy 1994). The Schur numbers also satisfy the inequality
and some positive constant 
(Abbott and Moser 1966, Abbott and Hanson
1972, Exoo 1994). Schur's Ramsey theorem
also shows that
is the 
-color
Ramsey number for a triangle
graph. The known Schur numbers are 1, 4, 13, 44, 160 (OEIS A045652).
The value of 
is due to Baumert (Golomb and Baumert 1965, Abbott and Hanson 1972). Exoo (1994)
established the lower bound 
, and Heule (2018) proved that 
. Fredricksen and Sweet (2000) established the lower
bounds 
and 
. Rowley (2021) improved
the latter to 
.
by giving an explicit partition
of 
into six sum-free
sets. Their construction also gives 
, 
, 
, and the Ramsey
number bound 
.
. It uses the least unavoidable endpoint, rather than
the largest avoidable endpoint used for 
(Khamseh 2020).
." 22 Oct 2026. 
." Elec. J. Combin. 1, R8, 1994. 
." J. Combin. Theory Ser. A 27,
376-377, 1979.Fredricksen, H. and Sweet, M. M. "Symmetric
Sum-Free Partitions and Lower Bounds for Schur Numbers." Elec. J. Combin. 7,
R32, 2000. 
." J. Math. 2020, 7069730, 2020. 
and Some Interesting Templates." 8 Jul 2021. 
(mod 
)." Jahresber. Deutsche Math.-Verein. 25,
114-116, 1916.Sloane, N. J. A. Sequences 
." Disc. Math. 4,
389-396, 1973.