ON A PROBLEM of ERDÅS about GRAPHS WHOSE SIZE IS the TURÁN NUMBER PLUS ONE

We consider finite simple graphs. Given a graph H and a positive integer the Turán number of H for the order denoted is the maximum size of a graph of order n not containing H as a subgraph. ErdÅ's asked: 'For which graphs H is it true that every graph on n vertices and edges contains at least two H's? Perhaps this is always true.' We solve this problem in the negative by proving that for every integer there exists a graph H of order k and at least two orders n such that there exists a graph of order n and size which contains exactly one copy of Denote by the -cycle. We also prove that for every integer n with there exists a graph of order n and size which contains exactly one copy of but, for or the minimum number of copies of in a graph of order n and size is two.