Schur Number
The Schur numbers are the numbers in the partitioning of a set which are guaranteed to exist by Schur's Lemma. Schurnumbers 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).See also Ramsey Number, Ramsey's Theorem, Schur's Lemma, Schur's Theorem
References
Frederickson, H. ``Schur Numbers and the Ramsey Numbers 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 A045652in ``The On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.
.'' J. Combin. Theory Ser. A 27, 376-377, 1979.
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