On the number of monochromatic Schur triples

Authors:
Boris A. Datskovsky
Affiliations:
Department of Mathematics, Temple University, Philadelphia, PA 19122, USA
Venue:
Advances in Applied Mathematics
Year:
2003

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Cited 4

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Abstract

Let S={1,2,...,n}, and let S=S"1@?S"2 be a partition of S in two disjoint subsets. A triple (a,b,c), a,b,c@?S, is called a Schur triple if a+b=c. If in addition a, b, and c all lie in the same subset S"i of S, we call the triple (a,b,c) monochromatic. In this paper we give a simple proof that the minimal number of monochromatic Schur triples is asymptotic to n^2/11. We also show that the number of monochromatic Schur triples modulo n equals n^2-3|S"1||S"2|.