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

for and some constant . Schur's Theorem also shows that

where is a Ramsey Number. The first few Schur numbers are 1, 4, 13, 44, , ... (Sloane's A045652).

**References**

Frederickson, H. ``Schur Numbers and the Ramsey Numbers
.'' *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.

© 1996-9

1999-05-26