Intersecting families of sets are typically trivial

TL;DR

This paper improves the understanding of the structure of large intersecting families of sets, showing that for sufficiently large n relative to k, almost all such families are trivial stars, using advanced combinatorial tools.

Contribution

It extends previous results by lowering the threshold for n, demonstrating that almost all large intersecting families are trivial stars for n ≥ 2k + 100ln k.

Findings

Almost all large intersecting families are trivial stars for n ≥ 2k + 100ln k.

The proof employs Sapozhenko's graph container lemma and the Das-Tran removal lemma.

The result improves previous bounds on n for the structure of intersecting families.

Abstract

A family of subsets of $[n]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran independently showed that for $n\geq 2k + c\sqrt{k\ln k}$, almost all $k$-uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for $n\geq 2k+ 100\ln k$. Our proof uses, among others, Sapozhenko's graph container lemma and the Das-Tran removal lemma.