On the random version of the Erd\H{o}s matching conjecture

TL;DR

This paper investigates the independence number of random Kneser hypergraphs, extending known results about the Erd ext{o}s-Ko-Rado theorem to hypergraphs with a probabilistic edge retention, especially when parameters are small.

Contribution

It generalizes previous results on the independence number of random graphs to the setting of random Kneser hypergraphs, proving the Erd ext{o}s matching conjecture's analogue for small probabilities.

Findings

The random Erd ext{o}s-Ko-Rado theorem holds for very small p.

The Erd ext{o}s matching conjecture is valid in the random hypergraph setting for small p.

The independence number remains predictable under random edge deletion in hypergraphs.

Abstract

The Kneser hypergraph ${\rm KG}^r_{n,k}$ is an $r$-uniform hypergraph with vertex set consisting of all $k$-subsets of $\{1,\ldots,n\}$ and any collection of $r$ vertices forms an edge if their corresponding $k$-sets are pairwise disjoint. The random Kneser hypergraph ${\rm KG}^r_{n,k}(p)$ is a spanning subhypergraph of ${\rm KG}^r_{n,k}$ in which each edge of ${\rm KG}^r_{n,k}$ is retained independently of each other with probability $p$. The independence number of random subgraphs of ${\rm KG}^2_{n,k}$ was recently addressed in a series of works by Bollob{\'a}s, Narayanan, and Raigorodskii (2016), Balogh, Bollob{\'a}s, and Narayanan (2015), Das and Tran (2016), and Devlin and Kahn (2016). It was proved that the random counterpart of the Erd\H{o}s-Ko-Rado theorem continues to be valid even for very small values of $p$. In this paper, generalizing this result, we will investigate the independence number of random Kneser hypergraphs ${\rm KG}^r_{n,k}(p)$. Broadly speaking, when $k$ is much smaller that $n$, we will prove that the random analogue of the Erd\H{o}s matching conjecture is true even for extremely small values of $p$.