Schur Number

The Schur numbers are the numbers in the partitioning of a set which are guaranteed to exist by Schur's Lemma. Schur numbers satisfy the inequality

$$s(k)\geq c(315)^{k/5}$$

for $k>5$ and some constant $c$. Schur's Theorem also shows that

$$s(n)\leq R(n),$$

where $R(n)$ is a Ramsey Number. The first few Schur numbers are 1, 4, 13, 44, $(\geq 157)$, ... (Sloane's A045652).

See also Ramsey Number, Ramsey's Theorem, Schur's Lemma, Schur's Theorem

References

Frederickson, H. ``Schur Numbers and the Ramsey Numbers $N(3, 3, \ldots, 3;2)$.'' J. Combin. Theory Ser. A 27, 376-377, 1979.

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.

Sloane, N. J. A. Sequence A045652 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.