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arXiv · 0806.2023

Shadows and intersections: stability and new proofs

Peter Keevash
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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.

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Peter Keevash. 2008-06-12. Shadows and intersections: stability and new proofs. https://doi.org/10.1016/j.aim.2008.03.023

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