Shadows and intersections: stability and new proofs

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• Peter Keevash (Queen Mary London)
• Thursday 14 May 2009, 14:30-15:30
• MR12.

If you have a question about this talk, please contact Andrew Thomason.

We give a short new proof of a version of the Kruskal-Katona theorem due to Lov\’asz. 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 Lov\’asz’s theorem that answers a question of Frankl and Tokushige.

This talk is part of the Combinatorics Seminar series.

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