Asymptotically optimal lower bounds on weak saturation numbers for hypergraphs

TL;DR

This paper extends known lower bounds on weak saturation numbers from graphs to hypergraphs, introducing a polymatroid-based method to establish their asymptotic optimality.

Contribution

It generalizes lower bounds for hypergraph weak saturation numbers and introduces a novel polymatroid-based approach for deriving these bounds.

Findings

Established asymptotically optimal lower bounds for hypergraph weak saturation numbers.

Generalized a linear algebraic method to handle non-integer asymptotic coefficients.

Extended known results from graphs to 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 $H$. For the case of graphs ($r = 2$), asymptotically optimal general lower bounds for these numbers in terms of the minimum vertex degree of $H$ are known. In this work, we generalize these bounds to the case of hypergraphs and establish their asymptotic optimality. To prove this, we introduce a lower bound method based on polymatroids. This method generalizes a linear algebraic method but, unlike the original version, makes it possible to derive lower bounds with non-integer asymptotic coefficients.