Color avoidance for monotone paths

TL;DR

This paper investigates the color-avoiding Ramsey numbers for monotone paths, revealing a significant reduction in tower height compared to traditional monochromatic cases, especially notable at uniformity three where the growth shifts from exponential to polynomial.

Contribution

It extends the understanding of hypergraph Ramsey numbers by analyzing color-avoiding variants, showing a lower tower height and a transition from exponential to polynomial growth at uniformity three.

Findings

Color-avoiding Ramsey numbers are lower than monochromatic ones.

At uniformity three, the growth of these numbers is polynomial.

The tower height decreases by one exponential in the color-avoiding setting.

Abstract

In 2014, Moshkovitz and Shapira determined the tower height for hypergraph Ramsey numbers of tight monotone paths. We address the color-avoiding version of this problem in which one no longer necessarily seeks a monochromatic subgraph, but rather one which avoids some colors. This problem was previously studied in uniformity two by Loh and by Gowers and Long. We show, in general, that the tower height for such Ramsey numbers requires one less exponential than in the usual setting. The transition occurs at uniformity three, where the usual Ramsey numbers of monotone paths of length $n$ are exponential in $n$, but the color-avoiding Ramsey numbers turn out to be polynomial.