On the measure of intersecting families, uniqueness and stability
 Ehud Friedgut
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Let t≥1 be an integer and let A be a family of subsets of {1,2,…,n} every two of which intersect in at least t elements. Identifying the sets with their characteristic vectors in {0,1}^{ n } we study the maximal measure of such a family under a non uniform product measure. We prove, for a certain range of parameters, that the tintersecting families of maximal measure are the families of all sets containing t fixed elements, and that the extremal examples are not only unique, but also stable: any tintersecting family that is close to attaining the maximal measure must in fact be close in structure to a genuine maximum family. This is stated precisely in Theorem 1.6.
We deduce some similar results for the more classical case of ErdősKoRado type theorems where all the sets in the family are restricted to be of a fixed size. See Corollary 1.7.
The main technique that we apply is spectral analysis of intersection matrices that encode the relevant combinatorial information concerning intersecting families. An interesting twist is that part of the linear algebra involved is done over certain polynomial rings and not in the traditional setting over the reals.
A crucial tool that we use is a recent result of Kindler and Safra [22] concerning Boolean functions whose Fourier transforms are concentrated on small sets.
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 Title
 On the measure of intersecting families, uniqueness and stability
 Journal

Combinatorica
Volume 28, Issue 5 , pp 503528
 Cover Date
 20080901
 DOI
 10.1007/s0049300823189
 Print ISSN
 02099683
 Online ISSN
 14396912
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 05D05
 05C69
 Industry Sectors
 Authors

 Ehud Friedgut ^{(1)}
 Author Affiliations

 1. Institute of Mathematics, Hebrew University, Jerusalem, Israel