The Ajtai-Erdos-Komlos-Szemeredi conjecture on independent sets in K_r-free graphs

Theorem 1.1: for every $r\ge4$ there is $c_r>0$ such that every finite $K_r$-free graph with $n$ vertices and average degree $d\ge2$ has $\alpha(G)\ge c_r n\log d/d$, removing the $\log\log d$ loss in Shearer's 1995 bound. The proof maximizes an entropy functional over vertex weights and shows that the weighted triangle mass is at most $O_r$ times the edge mass, via random multipliers, vertex splitting and a recursion on the clique size. The constant $c_r$ is not optimized and no leading coefficient (as in Shearer's triangle-free result) is claimed.

Theorem 1.1: for every $r\ge4$ there is $c_r>0$ such that every finite $K_r$-free graph with $n$ vertices and average degree $d\ge2$ has $\alpha(G)\ge c_r n\log d/d$, removing the $\log\log d$ loss in Shearer's 1995 bound. The proof maximizes an entropy functional over vertex weights and shows that the weighted triangle mass is at most $O_r$ times the edge mass, via random multipliers, vertex splitting and a recursion on the clique size. The constant $c_r$ is not optimized and no leading coefficient (as in Shearer's triangle-free result) is claimed.