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arXiv · 2104.11788

Lower Bound on the Size-Ramsey Number of Tight Paths

Christian Winter
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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 $n$, $R^{(k)}(P^{(k)}_{n})= \Omega\big(\log (k)n\big)$.

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Christian Winter. 2021-04-23. Lower Bound on the Size-Ramsey Number of Tight Paths. https://arxiv.org/abs/2104.11788

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