On "stability" in the Erd\H{o}s-Ko-Rado theorem

TL;DR

This paper investigates the stability of maximum independent sets in random subgraphs of Kneser graphs, establishing probabilistic thresholds and characterizing the structure of these sets for large parameters.

Contribution

It proves the existence of a fixed probability threshold where maximum independent sets are exactly the stars, and completes the understanding of this threshold for various parameters.

Findings

Existence of a fixed p<1 with high probability for the stability property

Characterization of maximum independent sets as stars for large k

Determination of the threshold order of magnitude for general n and k

Abstract

Denote by $K_p(n,k)$ the random subgraph of the usual Kneser graph $K(n,k)$ in which edges appear independently, each with probability $p$. Answering a question of Bollob\'as, Narayanan, and Raigorodskii,we show that there is a fixed $p<1$ such that a.s. (i.e., with probability tending to 1 as $k \to \infty$) the maximum independent sets of $K_p(2k+1, k)$ are precisely the sets $\{A\in V(K(2k+1,k)): x\in A\}$ ($x\in [2k+1]$). We also complete the determination of the order of magnitude of the "threshold" for the above property for general $k$ and $n\geq 2k+2$. This is new for $k\sim n/2$, while for smaller $k $ it is a recent result of Das and Tran.