Let L be a given family of so called prohibited graphs . Let ex (n, L denote the maximum number of edges a simple graph of order n can have without containing subgraphs from L . A typical extremal graph problem is to determine ex (n, L , or at least, find good bounds on it . Results asserting that for a given L there exists a much smaller L* 9 L for which ex (n, L --ex (n, L* will be called compactness results . The main purpose of this paper is to prove some compactness results for the case when L consists of cycles . One of our main tools will be finding lower bounds on the number of paths P1 + I in a graph of n vertices and E edges ., witch is, in fact, a "supersaturated" version of a wellknown theorem of Erdős and Gallai .
In REF , Erdős and Simonovits conjecture the following.
Compactness results in extremal graph theory
Combinatorica
Mathematics Computer Science