A short proof of Tuza's conjecture for weak saturation in hypergraphs

Abstract

Given an $r$-uniform hypergraph $H$ and a positive integer $n$, the weak saturation number $\mathrm{wsat}(n,H)$ is the minimum number of edges in an $r$-uniform hypergraph $F$ on $n$ vertices such that the missing edges in $F$ can be added, one at a time, so that each added edge creates a copy of $H$. Shapira and Tyomkyn (Proceedings of the American Mathematical Society, 2023) proved Tuza's conjecture on asymptotic behaviour of $\mathrm{wsat}(n, H)$. In this paper we provide a significantly shorter proof of the conjecture.