Fractional Turan's theorem and bounds for the chromatic number

Turan's theorem gives conditions for finding large complete subgraphs in a graph in terms of the number of edges. In this work we look for families of graphs in which there is a similar theorem, but which will allow us to find larger complete subgraphs. This question is motivated by a geometric result by Katchalski and Liu concerning a fractional Helly theorem. We present results in two directions. On the one hand, we characterize the families of graphs for which there exists a constant β such that a proportion of α edges guarantees the existence of a complete subgraph using a proportion of αβ of the total number of vertices. On the other hand, we give a criterion to guarantee that in a family of graphs any fixed proportion of the edges is sufficient to conclude that the order of the largest complete subgraph goes to infinity as the number of vertices goes to infinity. These results are obtained by giving an upper bound for the chromatic number of a graph in terms of the clique number and an arbitrarily small proportion of the number of vertices.