Integer colorings with no rainbow k-term arithmetic progression

• Published in: European journal of combinatorics Vol. 104; p. 103547
• Main Authors: Lin, Hao, Wang, Guanghui, Zhou, Wenling
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.08.2022
• ISSN: 0195-6698; 1095-9971
• DOI: 10.1016/j.ejc.2022.103547

In this paper, we study the rainbow Erdős–Rothschild problem with respect to k-term arithmetic progressions. For a set of positive integers S⊆[n], an r-coloring of S is rainbowk-AP-free if it contains no rainbow k-term arithmetic progression. Let gr,k(S) denote the number of rainbow k-AP-free r-colorings of S. For sufficiently large n and fixed integers r≥k≥3, we show that gr,k(S)<gr,k([n]) for any proper subset S⊂[n]. Further, we prove that limn→∞gr,k([n])/(k−1)n=rk−1. Our result is asymptotically best possible and implies that, almost all rainbow k-AP-free r-colorings of [n] use only k−1 colors.