Sub-Ramsey Numbers for Arithmetic Progressions and Schur Triples

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 different colors. We prove that 17 8 k +O(1) ≤ sr(3, k) ≤ 15 7 k + O(1) and sr(m, 2) > ⌊ 2 2 ⌋, 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 φ : [n] → [n] without a fixed point such that for every AP (m) A in [n], the set A∩φ(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 define 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) = ⌊ 5k 2 ⌋+ 1.