Stability anaylsis for k-wise intersecting families

TL;DR

This paper extends the Erd ext{"o}s-Ko-Rado theorem by establishing a stability version for k-wise intersecting families of r-subsets, using advanced combinatorial techniques involving Cayley graphs and expansion properties.

Contribution

It provides a stability result for a generalized intersecting family theorem, employing a novel adaptation of Katona's circle method and Cayley graph expansion analysis.

Findings

Proves a stability version of Frankl's generalization of Erd ext{"o}s-Ko-Rado theorem.

Uses Cayley graph expansion properties to analyze intersecting families.

Establishes bounds on the size of k-wise intersecting families under certain conditions.

Abstract

We consider the following generalization of the seminal Erd\H{o}s-Ko-Rado theorem, due to Frankl. For some k>=2, let F be a k-wise intersecting family of r-subsets of an n element set X, i.e. for any k sets F1,...,Fk in F, their intersection is nonempty. If r <= ((k-1)n)/k, then |F|<= {n-1 \choose r-1}. We prove a stability version of this theorem, analogous to similar results of Dinur-Friedgut, Keevash-Mubayi and others for the Erd\H{o}s-Ko-Rado theorem. The technique we use is a generalization of Katona's circle method, initially employed by Keevash, which uses expansion properties of a particular Cayley graph of the symmetric group.