On a class of degenerate extremal graph problems

Given a class ? of (so called “forbidden”) graphs, ex (n, ?) denotes the maximum number of edges a graphG ⁿ of ordern can have without containing subgraphs from ?. If ? contains bipartite graphs, then ex (n, ?)=O(n ^2?c ) for somec>0, and the above problem is calleddegenerate. One important degenerate extremal problem is the case whenC _2k , a cycle of 2k vertices, is forbidden. According to a theorem of P. Erd?s, generalized by A. J. Bondy and M. Simonovits 3₂, ex (n, {C _2k })=O(n ^1+1/k ). In this paper we shall generalize this result and investigate some related questions.