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

TL;DR

This paper investigates the stability of the Erdős-Ko-Rado theorem under random edge deletions in Kneser graphs, establishing a sharp threshold for when the independence number remains unchanged.

Contribution

It provides a sharp threshold result for the independence number of randomly edge-deleted Kneser graphs, extending the classical Erdős-Ko-Rado theorem to a probabilistic setting.

Findings

Identifies the probability regimes where the independence number is preserved

Establishes a sharp threshold for the stability of the independence number

Provides a probabilistic analogue of the Erdős-Ko-Rado theorem

Abstract

Delete the edges of a Kneser graph independently of each other with some probability: for what probabilities is the independence number of this random graph equal to the independence number of the Kneser graph itself? We prove a sharp threshold result for this question in certain regimes. Since an independent set in the Kneser graph is the same as a uniform intersecting family, this gives us a random analogue of the Erd\H{o}s-Ko-Rado theorem.