November 16
Henry Towsner,
University of Pennsylvania
Nonstandard analysis and new standard proof of the containers theorem
The hypergraph containers theorem, introduced by Balogh-Morris-Samotij and Saxton-Thomason, is a breakthrough in extremal combinatorics which gives a tool for controlling the number of independent sets in finite hypergraphs. (We do not assume any familiarity with the containers method!) The original proof depends on giving a program which constructs 'containers' for independent sets and analyzing its behavior to ensure the construction has the right properties. Inspired by the idea of pseudofinite dimension in ultraproducts, we give a new, elementary and nonalgorithmic, proof of the containers theorem which gives an direct characterization of the 'containers'. Joint work with Bernshteyn, Delcourt, and Tserunyan.