Schur (1916) proved that no matter how the set of positive integers less than or equal to 
(where 
is the floor function)
is partitioned into 
classes, one class must contain integers 
, 
,
such that 
, where 
and 
are not necessarily distinct. The least integer 
with this property is known as the
Schur number. The upper bound has since been slightly
improved to 
.
Schur's Problem
See also
Combinatorics, Ramsey Number, Schur Number, Schur's Partition Theorem, Sum-Free SetExplore with Wolfram|Alpha
References
Abbott, H. L. and Hanson, D. "A Problem of Schur and Its Generalizations." Acta Arith. 20, 175-187, 1972.Abbott, H. L. and Moser, L. "Sum-Free Sets of Integers." Acta Arith. 11, 393-396, 1966.Beutelspacher, A. and Brestovansky, W. "Generalized Schur Numbers." In Combinatorial Theory: Proceedings of a Conference Held at Schloss Rauischholzhausen, May 6-9, 1982 (Ed. D. Jungnickel and K. Vedder). Berlin: Springer-Verlag, pp. 30-38, 1982.Choi, S. L. G. "The Largest Sum-Free Subsequence from a Sequence of
Numbers." Proc. Amer. Math. Soc. 39, 42-44, 1973.Choi,
S. L. G.; Komlós, J.; and Szemerédi, R. "On Sum-Free
Subsequences." Trans. Amer. Math. Soc. 212, 307-313, 1975.Erdős,
P. "Some Problems and Results in Number Theory." In Number
Theory and Combinatorics: Japan 1984 (Ed. J. Akiyama). Singapore: World
Scientific, pp. 65-87, 1985.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.Irving, R. W. "An Extension of Schur's Theorem on Sum-Free
Partitions." Acta Arith. 25, 55-63, 1973.Schönheim,
J. "On Partitions of the Positive Integers with no 
, 
,
Belonging to Distinct Classes Satisfying
." In Number
Theory: Proceedings of the First Conference of the Canadian Number Theory Association
Held at the Banff Center, Banff, Alberta, April 17-27, 1988 (Ed. R. A. Mollin).
Berlin: de Gruyter, pp. 515-528, 1990.Wallis, W. D.; Street,
A. P.; and Wallis, J. S. Combinatorics:
Room Squares, Sum-free Sets, Hadamard Matrices. New York: Springer-Verlag,
1972.Referenced on Wolfram|Alpha
Schur's ProblemCite this as:
Weisstein, Eric W. "Schur's Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchursProblem.html