Lower bound on the size-Ramsey number of tight paths

Abstract

The size-Ramsey number $R^{(k)}(H)$ of a $k$-uniform hypergraph $H$ is the minimum number of edges in a $k$-uniform hypergraph $G$ with the property that each $2$-edge coloring of $G$ contains a monochromatic copy of $H$. For $k\ge2$ and $n\in\mathbb{N}$, a $k$-uniform tight path on $n$ vertices $P^{(k)}_{n,k-1}$ is defined as a $k$-uniform hypergraph on $n$ vertices for which there is an ordering of its vertices such that the edge set consists of all $k$-element intervals of consecutive vertices in this ordering. We show a lower bound on the size-Ramsey number of $k$-uniform tight paths, which is, considered assymptotically in both the uniformity $k$ and the number of vertices $n$, $R^{(k)}(P^{(k)}_{n,k-1})= \Omega_{n,k}\big(\log (k)n\big)$.