In 2001, Jeff Kahn showed that a disjoint union of $n/(2d)$ copies of the complete bipartite graph $K_{d,d}$ maximizes the number of independent sets over all $d$-regular bipartite graphs on n vertices, using Shearer’s entropy inequality. In this lecture I will mention several extensions and generalizations of this extremal result (to graphs and hypergraphs) and will describe a stability result (in the spectral sense) to Kahn’s result.

The lecture is based on joint works with Emma Cohen, David Galvin, Will Perkins, Michail Sarantis and Hiep Han.