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Counting independent sets in graphs

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, Łuczak, and Rödl.