Further applications of the Container Method

TL;DR

This paper explores advanced applications of the Container Method in combinatorics, demonstrating its effectiveness in counting specific graph structures, analyzing subgraph sizes in random graphs, and counting metric spaces, while also connecting to the Szemerédi Regularity Lemma.

Contribution

It extends the application of the Container Method to new combinatorial problems, including counting $C_4$-free graphs and analyzing metric spaces, showcasing its versatility.

Findings

Counted $C_4$-free graphs using the Container Method.

Analyzed the size of maximum $C_4$-free subgraphs in random graphs.

Counted metric spaces with a fixed number of points.

Abstract

Recently, Balogh--Morris--Samotij and Saxton--Thomason proved that hypergraphs satisfying some natural conditions have only few independent sets. Their main results already have several applications. However, the methods of proving these theorems are even more far reaching. The general idea is to describe some family of events, whose cardinality a priori could be large, only with a few certificates. Here, we show some applications of the methods, including counting $C_4$-free graphs, considering the size of a maximum $C_4$-free subgraph of a random graph and counting metric spaces with a given number of points. Additionally, we discuss some connections with the Szemer\'edi Regularity Lemma.