Sub-Ramsey numbers for arithmetic progressions

AbstractLetm ≥ 3 andk ≥ 1 be two given integers. Asub-k-coloring of [n] = {1, 2,...,n} is an assignment of colors to the numbers of [n] in which each color is used at mostk times. Call an $$S \subseteq [n]$$ arainbow set if no two of its elements have the same color. Thesub-k-Ramsey number sr(m, k) is defined as the minimumn such that every sub-k-coloring of [n] contains a rainbow arithmetic progression ofm terms. We prove thatΩ((k − 1)m2/logmk) ≤ sr(m, k) ≤ O((k − 1)m2 logmk) asm → ∞, and apply the same method to improve a previously known upper bound for a problem concerning mappings from [n] to [n] without fixed points.