Extremal Graphs without Cycles of Length 8 or Less

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