On Erd\H{o}s-Ko-Rado for random hypergraphs II

TL;DR

This paper investigates the Erdős-Ko-Rado property in random hypergraphs, showing that for certain parameters, these hypergraphs almost surely exhibit the property, extending classical combinatorial results to probabilistic settings.

Contribution

It establishes that random hypergraphs with high edge probability typically satisfy the Erdős-Ko-Rado property, answering a previously open question.

Findings

For n=2k+1 and p > 1 - ε, the hypergraph almost surely has the Erdős-Ko-Rado property.

The result holds as k approaches infinity, indicating a phase transition.

Mentions a similar probabilistic version of Sperner's Theorem.

Abstract

Denote by $\mathcal{H}_k (n,p)$ the random $k$-graph in which each $k$-subset of $\{1... n\}$ is present with probability $p$, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a fixed $\varepsilon >0$ such that if $n=2k+1$ and $p> 1-\varepsilon$, then w.h.p. (that is, with probability tending to 1 as $k\rightarrow \infty$), $\mathcal{H}_k (n,p)$ has the "Erd\H{o}s-Ko-Rado property." We also mention a similar random version of Sperner's Theorem.