Asymptotic enumeration via graph containers and entropy
Jinyoung Park
TL;DR
The paper surveys how graph container and entropy methods jointly yield sharp asymptotics for counting combinatorial objects such as antichains, independent sets, and graph homomorphisms. It presents a unifying framework: identify a tractable subfamily that captures the majority of the count and apply refined containers and entropy arguments to bound the remainder, enabling precise leading constants and even higher-order corrections in many cases. Key achievements include sharp asymptotics for the number of independent sets in the hypercube, the number of 4-colorings of the hypercube, Z-homomorphisms from Q_n, and, more broadly, asymptotics for antichains in both the Boolean lattice and irregular grids [t]^n. The work further extends container-based techniques to irregular settings and combines them with cluster expansion and isoperimetric inequalities, yielding breakthroughs in Ramsey-type problems and Lipschitz function ranges on expanders. Overall, the article highlights a powerful, versatile toolkit for asymptotic enumeration across diverse graph- and lattice-based combinatorial settings with wide theoretical and potential applied implications.
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.