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 sr(3, k) = 17 8 k + O(1) and, for m > 1 and k > 1, that sr(m, k) = Ω(m2 k), improving the previous bounds of Alon, Caro, and Tuza from 1989. Our new lower bound on sr(m, 2) immediately implies that for n ≤ m2 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...

• Type: Journal
• Year: 2007
• Where: ENDM
• Authors: Jacob Fox, Veselin Jungic, Rados Radoicic
• Keywords: ENDM 2007, M-term Arithmetic Progression, Minimal Positive Integer, Positive Integer