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 every `$2$-edge coloring' of $G$ contains a monochromatic copy of $H$. For $k\ge 2$ and $n\in \mathbb{N}$, a $k$-uniform tight path on $n$ vertices $P_n^{(k)}$ is defined as a $k$-uniform hypergraph on $n$ vertices for which there is an ordering of its vertices such that the edges are all sets of $k$ consecutive vertices with respect to this order. We prove 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 $n$, $R^{(k)}(P_n^{(k)})=\Omega (\log (k)n)$.

Accepted for JOC, 7 pages, 1 figure