A lower bound on the multicolor size-Ramsey numbers of paths in hypergraphs

TL;DR

This paper establishes new lower bounds on the multicolor size-Ramsey numbers of paths in hypergraphs, extending known results from graphs to hypergraphs and providing precise estimates for short paths.

Contribution

It provides the first general lower bounds for the multicolor size-Ramsey numbers of hypergraph paths, including tight bounds for short paths and extensions to overlapping paths.

Findings

Established a lower bound of _k(r^k n) for hypergraph paths

Determined the order of magnitude for short tight paths as _k(r^m)

Extended results to _k(r^2 n) for _k(r^2  n) in overlapping paths

Abstract

The $r$-color size-Ramsey number of a $k$-uniform hypergraph $H$, denoted by $\hat{R}_r(H)$, is the minimum number of edges in a $k$-uniform hypergraph $G$ such that for every $r$-coloring of the edges of $G$ there exists a monochromatic copy of $H$. In the case of $2$-uniform paths $P_n$, it is known that $\Omega(r^2n)=\hat{R}_r(P_n)=O((r^2\log r)n)$ with the best bounds essentially due to Krivelevich. In a recent breakthrough result, Letzter, Pokrovskiy, and Yepremyan gave a linear upper bound on the $r$-color size-Ramsey number of the $k$-uniform tight path $P_{n}^{(k)}$; i.e. $\hat{R}_r(P_{n}^{(k)})=O_{r,k}(n)$. Winter gave the first non-trivial lower bounds on the 2-color size-Ramsey number of $P_{n}^{(k)}$ for $k\geq 3$; i.e. $\hat{R}_2(P_{n}^{(3)})\geq \frac{8}{3}n-O(1)$ and $\hat{R}_2(P_{n}^{(k)})\geq \lceil\log_2(k+1)\rceil n-O_k(1)$ for $k\geq 4$. We consider the problem of giving a lower bound on the $r$-color size-Ramsey number of $P_{n}^{(k)}$ (for fixed $k$ and growing $r$). Our main result is that $\hat{R}_r(P_n^{(k)})=\Omega_k(r^kn)$ which generalizes the best known lower bound for graphs mentioned above. One of the key elements of our proof is a determination of the correct order of magnitude of the $r$-color size-Ramsey number of every sufficiently short tight path; i.e. $\hat{R}_r(P_{k+m}^{(k)})=\Theta_k(r^m)$ for all $1\leq m\leq k$. All of our results generalize to $\ell$-overlapping $k$-uniform paths $P_{n}^{(k, \ell)}$. In particular we note that when $1\leq \ell\leq \frac{k}{2}$, we have $\Omega_k(r^{2}n)=\hat{R}_r(P_{n}^{(k, \ell)})=O((r^2\log r)n)$ which essentially matches the best known bounds for graphs mentioned above. Additionally, in the case $k=3$, $\ell=2$, and $r=2$, we give a more precise estimate which implies $\hat{R}_2(P^{(3)}_{n})\geq \frac{28}{9}n-O(1)$, improving on the above-mentioned lower bound of Winter in the case $k=3$.