Some new upper bounds of ex(n; {C3,C4})

The extremal number ex(n;{C₃,C₄}) or simply ex(n;4) denotes the maximal number of edges in a graph on n vertices with forbidden subgraphs C₃ and C₄. The exact number of ex(n;4) is only known for n up to 32 and n=50. There are upper and lower bounds of ex(n;4) for other values of n. In this paper, we improve the upper bound of ex(n;4) for n=33,34,...,42 and also n=d²+1 for any positive integer d≠7,57.