# Mini-workshop on Graph Container methods

Date&Time: May 4th 9:30AM-1:30PM EST (1:30PM-5:30PM UTC, 2:30PM-6:30PM Central European Time)

Description: Graph Container methods are a powerful set of combinatorial techniques that continue to see a lot of use. The goal of this workshop is to focus on some very recent applications of these methods, and in particular, of Sapozhenko's Graph Container Lemma''.

All times given below are on May 4th in EST.

9:30AM - 10:15AM : Introduction to graph containers - Aditya Potukuchi

Abstract: I will explain the definitions, motivation, and basic details of graph containers and in paticular, Sapozhenko's container lemma. If time permits, I will provide brief sketches of the proofs as well. No prior knowledge is assumed.

10:20AM - 11:05AM : The number of balanced independent sets in the Hamming cube - Jinyoung Park

Abstract: An independent set in the Hamming cube is called balanced if it contains the same number of even and odd vertices. In this talk, we will discuss the number of balanced independent sets in the Hamming cube and its relation to Sapozhenko's graph container lemma.

11:20AM - 12:05AM :The typical structure of intersecting families - Lina Li

Abstract: A family of subsets of [n] is intersecting if every pair of its members has a non-trivial intersection. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran independently showed that for n>= 2k + c\sqrt{k\ln k}, almost all k-uniform intersecting families are stars. Significantly improving their results, we show that the same conclusion holds for n>= 2k+ 100\ln k. Our proof uses the Sapozhenko’s graph container method and the Das-Tran removal lemma.

This is joint work with Jòzsef Balogh, Ramon I. Garcia and Adam Zsolt Wagner.

12:10PM - 12:55PM : Enumerating independent sets in Abelian Cayley graphs - Liana Yepremyan

Abstract: We show that any Cayley graph on an Abelian group of order 2n and degree \tilde{\Omega}(\log n) has at most 2^{n+1}(1 + o(1)) independent sets. This bound is tight up to to the o(1) term whenever the graph is bipartite. Proof techniques include the graph container method of Sapozhenko and the Plünnecke-Rusza-Petridis inequality from additive combinatorics.

This is joint work with Aditya Potukuchi

12:55PM - 1:30PM : Informal discussion