On the Density of Subgraphs in a Graph with Bounded Independence Number

Letσ(n, m, k) be the largest numberσ∈[0, 1] such that any graph onnvertices with independence number at mostmhas a subgraph onkvertices with at lest [formula]edges. Up to a constant multiplicative factor, we determineσ(n, m, k) for alln,emsp14;m, k. For logn⩽m=k⩽n, our result givesσ(n, m, m)=Θ(log(n/m)/m), which was conjectured by Alon (Random Structures Algorithms9(1996), 271–278).