Maximizing the Number of Independent Sets of a Fixed Size

Let i_t (G) be the number of independent sets of size t in a graph G. Engbers and Galvin asked how large i_t (G) could be in graphs with minimum degree at least δ. They further conjectured that when n ≥ 2δ and t ≥ 3, i_t (G) is maximized by the complete bipartite graph K_δ,n−δ . This conjecture has recently drawn the attention of many researchers. In this short note, we prove this conjecture.