Lower Bound on the Size-Ramsey Number of Tight Paths

TL;DR

This paper establishes a lower bound on the size-Ramsey number of k-uniform tight paths, showing it grows at least logarithmically with the number of vertices and the uniformity, advancing understanding of hypergraph Ramsey theory.

Contribution

It provides the first asymptotic lower bound on the size-Ramsey number of k-uniform tight paths, linking the bound to both uniformity and path length.

Findings

Lower bound of (log(k)n) on the size-Ramsey number

Asymptotic growth rate depends on uniformity and number of vertices

Advances theoretical understanding of hypergraph Ramsey numbers

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 $k\ge2$ and $n\in\mathbb{N}$, a $k$-uniform tight path on $n$ vertices $P^{(k)}_{n}$ 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 $n$, $R^{(k)}(P^{(k)}_{n})= \Omega\big(\log (k)n\big)$.