Extremal Graphs without Cycles of Length 8 or Less

Let ex(n;t) denote the maximum number of edges in a graph G having order n without cycles of length t or less. We prove ex(23;8)=28,ex(24;8)=20 and ex(25;8)=30. Furthermore, we present new lower and upper bounds for n⩽49 and the extremal numbers when known.