Size of Graphs with High Girth

Let n⩾4 be a positive integer and let ex(ν;{C3,…,Cn}) denote the maximum number of edges in a {C3,…,Cn}-free simple graph of order ν. This paper gives the exact value of this function for all ν up to ⌊(16n−15)/5⌋. This result allows us to deduce all the different values of the girths that such extremal graphs can have.Let k⩾0 be an integer. For each n⩾2log2(k+2) there exists ν such that every extremal graph G with e(G)−ν(G)=k has minimal degree at most 2, and is obtained by adding vertices of degree 1 and/or by subdividing a graph or a multigraph H with δ(H)⩾3 and e(H)−ν(H)=k.