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 $p$.
Categories: math.CO