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

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arxiv 2504.03816 v3 pith:7DXRJKWN submitted 2025-04-04 math.CO

classification math.CO
keywords conjectureaddededgeshypergraphmathrmnumberproofsaturation
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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.

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  1. When does a tree activate the random graph?

    math.CO 2025-07 accept novelty 8.0 of 10

    The critical probability for the existence of a K3-activating spanning tree in G(n,p) is p = n^{-1/3-o(1)}.

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