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arXiv · 1907.08086

The size-Ramsey number of 3-uniform tight paths

Jie Han·Yoshiharu Kohayakawa·Shoham Letzter·Guilherme Oliveira Mota·Olaf Parczyk
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Abstract

Given a hypergraph $H$, the size-Ramsey number $\hat{r}_2(H)$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that in any colouring of the edges of $G$ with two colours there is a monochromatic copy of $H$. We prove that the size-Ramsey number of the $3$-uniform tight path on $n$ vertices $P^{(3)}_n$ is linear in $n$, i.e., $\hat{r}_2(P^{(3)}_n) = O(n)$. This answers a question by Dudek, Fleur, Mubayi, and Rödl for $3$-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417-434], who proved $\hat{r}_2(P^{(3)}_n) = O(n^{3/2} \log^{3/2} n)$.

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Jie Han, Yoshiharu Kohayakawa, Shoham Letzter, Guilherme Oliveira Mota, Olaf Parczyk. 2021-06-07. The size-Ramsey number of 3-uniform tight paths. https://doi.org/10.19086/aic.24581

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