Elsevier

Advances in Mathematics

Volume 218, Issue 5, 1 August 2008, Pages 1685-1703
Advances in Mathematics

Shadows and intersections: Stability and new proofs

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Abstract

We give a short new proof of a version of the Kruskal–Katona theorem due to Lovász. 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 Lovász's theorem that answers a question of Frankl and Tokushige.

Keywords

Set systems
Shadows
Intersections
Stability

Cited by (0)

Research supported in part by NSF grant DMS-0555755.