We investigate the values of t(n), the maximum number of edges in a graph with n vertices and not containing a four-cycle. Techniques for finding these are developed and the values of t(n) for all n up to 21 are obtained. All the corresponding extremal graphs are found.

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A method is described of constructing a class of self‐complementary graphs, that includes a self‐complementary graph, containing no K5, with 41 vertices and a self‐complementary graph, containing no K7, with 113 vertices. The latter construction gives the improved Ramsey number lower bound r(7, 7) ≥ 114.

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