Removal and Stability for Erd\H{o}s-Ko-Rado

TL;DR

This paper proves a simple removal lemma for large set families with few disjoint pairs, showing they are close to unions of stars, and applies it to determine thresholds for independence numbers in random Kneser graphs.

Contribution

It provides a new, simple proof of a removal lemma for large families and applies it to analyze independence numbers in random Kneser graphs.

Findings

Families close to unions of stars with few disjoint pairs

Sharp thresholds for independence number in random Kneser graphs

Bounds on critical probability for certain parameters

Abstract

A $k$-uniform family of subsets of $[n]$ is intersecting if it does not contain a disjoint pair of sets. The study of intersecting families is central to extremal set theory, dating back to the seminal Erd\H{o}s-Ko-Rado theorem of 1961 that bounds the size of the largest such families. A recent trend has been to investigate the structure of set families with few disjoint pairs. Friedgut and Regev proved a general removal lemma, showing that when $\gamma n \le k \le (\tfrac12 - \gamma)n$, a set family with few disjoint pairs can be made intersecting by removing few sets. We provide a simple proof of a removal lemma for large families, showing that families of size close to $\ell \binom{n-1}{k-1}$ with relatively few disjoint pairs must be close to a union of $\ell$ stars. Our lemma holds for a wide range of uniformities; in particular, when $\ell = 1$, the result holds for all $2 \le k < \frac{n}{2}$ and provides sharp quantitative estimates. We use this removal lemma to settle a question of Bollob\'as, Narayanan and Raigorodskii regarding the independence number of random subgraphs of the Kneser graph $K(n,k)$. The Erd\H{o}s-Ko-Rado theorem shows $\alpha(K(n,k)) = \binom{n-1}{k-1}$. For some constant $c > 0$ and $k \le cn$, we determine the sharp threshold for when this equality holds for random subgraphs of $K(n,k)$, and provide strong bounds on the critical probability for $k \le \tfrac12 (n-3)$.