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arXiv · 2103.01942

Size-Ramsey numbers of powers of hypergraph trees and long subdivisions

Shoham Letzter·Alexey Pokrovskiy·Liana Yepremyan
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Abstract

The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $t$-power of the $r$-uniform tight path on $n$ vertices is linear in $n$, for every fixed $r, s, t$, thus answering a question of Dudek, La Fleur, Mubayi, and R\"odl (2017). In fact, we prove a stronger result that allows us to deduce that powers of bounded degree hypergraph trees and powers of `long subdivisions' of bounded degree hypergraphs have size-Ramsey numbers that are linear in the number of vertices. This extends and strongly generalises recent results about the linearity of size-Ramsey numbers of powers of bounded degree trees and of long subdivisions of bounded degree graphs.

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Shoham Letzter, Alexey Pokrovskiy, Liana Yepremyan. 2021-03-02. Size-Ramsey numbers of powers of hypergraph trees and long subdivisions. https://arxiv.org/abs/2103.01942

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