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 $\mathrm{\mid }N(S)\mathrm{\mid }\mathrm{/}\mathrm{\mid }S\mathrm{\mid }$ 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 $k\ge 6$, confirming Brandt’s conjecture; the bound is sharp for complete $(k-1)$-partite graphs with equal nonempty parts. We first prove the case $k=6$ and then extend it to all $k\ge 6$ by induction.