A sharp bound on the binding number of $K_k$-free graphs and the Brandt conjecture

The binding number of a graph is the smallest ratio $|N(S)|/|S|$ among all nonempty vertex sets $S$ whose neighborhood $N(S)$ is not the whole vertex set. We prove that every finite nonempty $K_k$-free graph has binding number at most $k-2$ for every $k\ge6$, confirming Brandt's conjecture; the boun