On extremal numbers of the triangle plus the four-cycle

For a family F of graphs, let ex(n,F) denote the maximum number of edges in an n -vertex graph which contains none of the members of F as a subgraph. A longstanding problem in extremal graph theory asks to determine the function ex(n,₃,C₄\) . Here we give a new construction for dense graphs of girth at least five with arbitrary number of vertices, providing the first improvement on the lower bound of ex(n,₃,C₄\) since 1976. As a corollary, this yields a negative answer to a problem in Chung-Graham [3].