Sub-Ramsey Numbers for Arithmetic Progressions and Schur Triples

Abstract

For a given positive integer k, sr(m; k) denotes the minimal positive integer such that every coloring of [n], n sr(m; k), that uses each color at most k times, yields a rainbow AP (m); that is, an m-term arithmetic progression, all of whose terms receive di erent colors. We prove that sr(3; k) = 17 8 k + O(1) and, for m > 1 and k > 1, that sr(m; k) = (mk), improving the previous bounds of Alon, Caro, and Tuza from 1989. Our new lower bound on sr(m; 2) immediately implies that for n m 2 2 , there exists a mapping : [n]! [n] without a xed point such that for every AP (m) in [n], the set A (A) is not empty. We also propose the study of sub-Ramsey{type problems for linear equations other than x + y = 2z. For a given positive integer k, we de ne ss(k) to be the minimal positive integer n such that every coloring of [n], n ss(k), that uses each color at most k times, yields a rainbow solution to the Schur equation x+ y = z. We prove that ss(k) = b 5k 2 c+ 1.