Schur's problem concerns the least interval length forcing a monochromatic solution to 
under every coloring with a fixed
number of colors. 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 
, where 
is 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
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Numbers." Proc. Amer. Math. Soc. 39, 42-44, 1973.Choi,
S. L. G.; Komlós, J.; and Szemerédi, R. "On Sum-Free
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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, Germany: 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 Resource. https://mathworld.wolfram.com/SchursProblem.html