Shadows and intersections: stability and new proofs

19 January 2010
Peter Keevash
<span hover_container="show_item_20226418" id="content_item_20226418" class="content hover_target"><span hover_container="show_item_20226418" class="commentable_icon_position_reference">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.</span></span>
  • Combinatorial Theory Seminar