Counting independent sets in graphs

TL;DR

This paper surveys an elementary yet powerful method for counting independent sets in graphs, illustrating its applications to various combinatorial problems and providing bounds and proofs for several graph and set properties.

Contribution

It presents a clear exposition of a classical enumeration method and demonstrates its versatility through multiple new applications and bounds in combinatorics.

Findings

Bounds on the number of independent sets in regular graphs

Results on sum-free subsets of integers

Analysis of $C_4$-free graphs and sparse random sets

Abstract

In this short survey article, we present an elementary, yet quite powerful, method of enumerating independent sets in graphs. This method was first employed more than three decades ago by Kleitman and Winston and has subsequently been used numerous times by many researchers in various contexts. Our presentation of the method is illustrated with several applications of it to `real-life' combinatorial problems. In particular, we derive bounds on the number of independent sets in regular graphs, sum-free subsets of $\{1, \ldots, n\}$, and $C_4$-free graphs and give a short proof of an analogue of Roth's theorem on $3$-term arithmetic progressions in sparse random sets of integers which was originally formulated and proved by Kohayakawa, \L uczak, and R\"odl.