Dedicated to the memory of Paul Erdős
A graph is called \(\)-free if it contains no cycle of length four as an induced subgraph. We prove that if a \(\)-free graph has n vertices and at least \(\) edges then it has a complete subgraph of \(\) vertices, where \(\) depends only on \(\). We also give estimates on \(\) and show that a similar result does not hold for H-free graphs––unless H is an induced subgraph of \(\). The best value of \(\) is determined for chordal graphs.
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