A short proof of Tuza's conjecture for weak saturation in hypergraphs
Nikolai Terekhov
TL;DR
The paper addresses the asymptotic behaviour of the weak saturation number $\mathrm{wsat}(n,H)$ for $r$-uniform hypergraphs by proving Tuza's conjecture with a shorter argument. It leverages Rödl's covering-type theorem to assemble small $\mathrm{wSAT}$ constructions into a large one on $[n]$ and develops a direct induction to show any remaining missing edge completes a copy of $H$, all while avoiding the $T_{r,h,s}$-template saturation technique. The main result establishes the existence of the limit $\lim_{n\to\infty} \mathrm{wsat}(n,H)/\binom{n-v}{s-1}$ (where $v=|V(H)|$ and $s=s(H)$) and yields the asymptotic form $\mathrm{wsat}(n,H)=C_H\binom{n-v}{s-1}(1+o(1))$ with a positive constant $C_H$, thereby confirming Tuza's conjecture in a streamlined fashion. This simplification enhances understanding of the growth rate of weak saturation in hypergraphs and provides a more accessible route to the asymptotic characterization of $\mathrm{wsat}(n,H)$.
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.