Asymptotic enumeration via graph containers and entropy

TL;DR

This paper surveys recent advances combining graph container methods and entropy to improve asymptotic enumeration of graph homomorphisms, highlighting their powerful applications in combinatorics.

Contribution

It introduces a novel approach that merges specialized container techniques with entropy to enhance enumeration results for graph homomorphisms.

Findings

Improved bounds on the number of graph homomorphisms

New insights into enumeration via combined methods

Enhanced understanding of extremal combinatorial structures

Abstract

The container methods are powerful tools to bound the number of independent sets of graphs and hypergraphs, and they have been extremely influential in the area of extremal and probabilistic combinatorics. We will focus on more specialized graph container methods due to Sapozhenko (1987) that deal with sets in expander graphs. Entropy, first introduced by Shannon (1948) in the area of information theory, is a measure of the expected amount of information contained in a random variable. Entropy has seen lots of fascinating applications in a wide range of enumeration problems. In this survey article, we will discuss recent developments that exploit a combination of the two methods on enumerating graph homomorphisms.