Shadows and intersections: stability and new proofs

TL;DR

This paper presents new combinatorial and algebraic proofs of stability and intersection theorems, extending classical results with novel methods and answering longstanding open questions.

Contribution

It introduces a new combinatorial proof of Lovász's version of the Kruskal-Katona theorem and extends it to stability results, also providing algebraic perspectives using Cayley graph expansion.

Findings

New combinatorial proof of Lovász's Kruskal-Katona theorem

Stability results describing near-extremal configurations

Algebraic proof using Cayley graph expansion

Abstract

We give a short new proof of a version of the Kruskal-Katona theorem due to Lov\'asz. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a question of Mubayi. This in turn leads to another combinatorial proof of a stability theorem for intersecting families, which was originally obtained by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi by means of a purely combinatorial result of Frankl. We also give an algebraic perspective on these problems, giving yet another proof of intersection stability that relies on expansion of a certain Cayley graph of the symmetric group, and an algebraic generalisation of Lov\'asz's theorem that answers a question of Frankl and Tokushige.