On the Matching Problem in Random Hypergraphs

TL;DR

This paper investigates the maximum size of sub-hypergraphs with a given matching number in random hypergraphs, showing that trivial sub-hypergraphs are optimal under certain conditions.

Contribution

It establishes that in certain regimes, the largest sub-hypergraphs with a fixed matching number are trivial, extending understanding of matchings in random hypergraphs.

Findings

Trivial sub-hypergraphs maximize size for fixed matching number

Results hold when n is much larger than k^2 s and p is sufficiently large

High probability results for the Erdős-Rényi random hypergraph model

Abstract

We study a variant of the Erd\H{o}s Matching Problem in random hypergraphs. Let $\mathcal{K}_p(n,k)$ denote the Erd\H{o}s-R\'enyi random $k$-uniform hypergraph on $n$ vertices where each possible edge is included with probability $p$. We show that when $n\gg k^{2}s$ and $p$ is not too small, with high probability, the maximum number of edges in a sub-hypergraph of $\mathcal{K}_p(n,k)$ with matching number $s$ is obtained by the trivial sub-hypergraphs, i.e. the sub-hypergraph consisting of all edges containing at least one vertex in a fixed set of $s$ vertices.