An Improved Upper Bound for Colorings Without Symmetrically Colored $k$-Term Arithmetic Progressions

Abstract

Given a coloring $c$ and an even $k\ge 4$, a nontrivial $k$-term arithmetic progression~($k$-AP) $a,a+d,\ldots,a+(k-1)d$ is called symmetrically colored if $c(a+(i-1)d)=c(a+(k-i)d)$, $\forall i\in[k/2]$. Deng, Tidor, and Zhao asked whether $[N]$ admits a coloring with $N^{o(1)}$ colors and no such 4-APs, and gave an $O(N^{\log_{22}3})$-coloring of $[N]$. We give an $O_k(p)$-coloring of $\mathbb Z/p^{k^2/4}\mathbb Z$ without such $k$-APs for every even $k\ge 4$ and every prime $p>k$, and hence an $O_k(N^{4/k^2})$-coloring of $[N]$, improving the exponent in the upper bound for $4$-APs from $\log_{22}3$ to $1/4$. The construction combines a carry-control coloring of base-$p$ digits with a layered field norm mapping. Together with Behrend-style product colorings, our result for $4$-APs gives $h(N)\leq N^{1/4+o(1)}$ in Erdős's Problem~160 on coloring every nontrivial 4-AP with at least three colors. This result also yields $ρ_4(α)=O_\varepsilon(α^{5-\varepsilon})$ for every $\varepsilon>0$, improving the bound toward Ruzsa's question. Our result for $k$-APs disproves Gowers' conjectured lower bound for all even $k\ge6$ for the first time.