Shadows and intersections: Stability and new proofs

• Autores: Peter Keevash
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 218, Nº 5, 2008, págs. 1685-1703
• Idioma: inglés
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• Resumen

  • 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