Girth of {C3, ..., Cs}-free extremal graphs

Let G be a {C"3,...,C"s}-free graph with as many edges as possible. In this paper we consider a question studied by several authors, the compulsory existence of the cycle C"s"+"1 in G. The answer has already been proved to be affirmative for s=3,4,5,6. In this work we show that the girth of G is g(G)=s+1 when the order of G is at least 1+s(s-22)^s^-^2-4s-4 if s is even, and 1+(s-1)^3((s-2)^2-14)^s^-^3^2-8s2(s-2)^2-10 if s is odd. This bound is an improvement of the best general result so far known. Moreover, we also prove in the case s=7 that the girth is g(G)=8 for order at least 14 and characterize all the extremal graphs whose girth is not 8.