On unavoidable graphs

How many edges can be in a graph which is forced to be contained in every graph on n verti-ces and e edges? 1n this paper we obtain bounds which are in many cases asymptotically best possible. A well-known theorem of Turán [10,11] asserts that every graph on n vertices and e edges must contain a complete subgraph on m vertices if e~2(m 1)(n-rL)+(2) where r satisfies r-n (mod m-1) and In this paper we consider a related extremal problem. A graph which is forced to be contained in every graph on n vertices and e edges is called an (n, e)-unavoidable graph, or in short, an unavoidable graph. Let f(n, e) denote the largest integer m with the property that there exists an (n, e)-unavoidable graph on ni edges. In this paper we prove the following : (~) fl n, e) = I if e-Il2 n .