An Extremal Problem For Random Graphs And The Number Of Graphs With Large Even-Girth

We study the maximal number of edges a C2k-free subgraph of a random graph Gn;p may have, obtaining best possible results for a range of p = p(n). Our estimates strengthen previous bounds of F uredi 12] and Haxell, Kohayakawa, and Luczak 13]. Two main tools are used here: the rst one is an upper bound for the number of graphs with large even-girth, i.e., graphs without short even cycles, with a given number of vertices and edges, and satisfying a certain additional pseudorandom condition; the second tool is the powerful result of Ajtai, Komll os, Pintz, Spencer, and Szemer edi 1] on uncrowded hypergraphs as given by Duke, Lefmann, and RR odl 7]. Part of the revision was carried out with the support of PROBRAL project 026/95, a CAPES{DAAD exchange programme, and PRONEX project 107/97.