Tower heights for color-avoiding Ramsey numbers of monotone paths

Abstract

Ramsey numbers of monotone paths in ordered hypergraphs form a natural higher-uniformity extension of the classical Erd\H{o}s--Szekeres theorems, and their tower height was determined by Moshkovitz and Shapira. A color-avoiding variant, initiated by Loh and further developed by Gowers and Long and by Mulrenin, Pohoata, and Zakharov, asks for monotone paths whose edges use only a bounded number of colors rather than a single color. For integers $q>p$, let $A_k(n;q,p)$ be the least integer $N$ such that every $q$-coloring of the ordered complete $k$-uniform hypergraph on $\{1,\ldots,N\}$ contains a monotone path of length $n$ whose edges use at most $p$ colors. We prove that, for every fixed $p$ and all sufficiently large $q$, the exact tower height of $A_k(n;q,p)$ is $\lceil (k-1)/p\rceil$. Thus the number of colors allowed on the path affects the Ramsey number at the level of tower height: allowing $p$ colors lowers the height from $k-1$ in the monochromatic problem to $\lceil (k-1)/p\rceil$. This answers questions of Mulrenin, Pohoata, and Zakharov. The upper bound follows from a simple block-compression argument. The main contribution is the matching lower bound, for which we develop a novel variant of the stepping-up method. A surprising feature of the proof is the appearance of the Morse--Hedlund theorem, a foundational result in symbolic dynamics and combinatorics on words. We establish and use a finite version of this theorem, which may be of independent interest.