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On the measure of intersecting families, uniqueness and stability. (English) Zbl 1199.05319
Summary: Let \(t\geq 1\) be an integer and let \(\mathcal A\) be a family of subsets of \(\{1,2,\dots, n\}\) 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 \(t\)-intersecting 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 \(t\)-intersecting 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 Erdős-Ko-Rado 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.

MSC:
05D05 Extremal set theory
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
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