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 rainbow k-
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
lim n→∞ gr, k ([n])/(k− 1) n= rk− 1. Our result is asymptotically best possible and implies that …

In this paper, we study the rainbow Erd\H {o} s-Rothschild problem with respect to 3-term
arithmetic progressions. We obtain the asymptotic number of $ r $-colorings of $[n] $ without
rainbow 3-term arithmetic progressions, and we show that the typical colorings with this
property are 2-colorings. We also prove that $[n] $ attains the maximum number of rainbow 3-
term arithmetic progression-free $ r $-colorings among all subsets of $[n] $. Moreover, the
exact number of rainbow 3-term arithmetic progression-free $ r $-colorings of $\mathbb {Z} …