On the size-Ramsey number of tight paths

TL;DR

This paper improves bounds on the size-Ramsey number of tight paths in hypergraphs for multiple colors, showing it is polynomial in the number of vertices and colors, which advances understanding of hypergraph Ramsey theory.

Contribution

The authors establish a new upper bound on the size-Ramsey number of tight paths in hypergraphs for any number of colors, generalizing previous results.

Findings

New upper bound: $oxed{ ext{O}(r^k (n ext{log} n)^{k/2})}$ for all $k ext{ and } r$.

Improved understanding of hypergraph Ramsey numbers for tight paths.

Generalization from 2-color to r-color case.

Abstract

For any $r\geq 2$ and $k\geq 3$, the $r$-color size-Ramsey number $\hat R(\mathcal{G},r)$ of a $k$-uniform hypergraph $\mathcal{G}$ is the smallest integer $m$ such that there exists a $k$-uniform hypergraph $\mathcal{H}$ on $m$ edges such that any coloring of the edges of $\mathcal{H}$ with $r$ colors yields a monochromatic copy of $\mathcal{G}$. Let $\mathcal{P}_{n,k-1}^{(k)}$ denote the $k$-uniform tight path on $n$ vertices. Dudek, Fleur, Mubayi and R\H{o}dl showed that the size-Ramsey number of tight paths $\hat R(\mathcal{P}_{n,k-1}^{(k)}, 2) = O(n^{k-1-\alpha} (\log n)^{1+\alpha})$ where $\alpha = \frac{k-2}{\binom{k-1}{2}+1}$. In this paper, we improve their bound by showing that $\hat R(\mathcal{P}_{n,k-1}^{(k)}, r) = O(r^k (n\log n)^{k/2})$ for all $k\geq 3$ and $r\geq 2$.