## Abstract

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*⫅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*P* ^{k+1} in a graph of*n* vertices and*E* edges., witch is, in fact, a “super-saturated” version of a wellknown theorem of Erdős and Gallai.

## AMS subject classification (1980)

05 C 35## Preview

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