Abstract
To everyk-graphG letπ(G) be the minimal real numberπ such that for everyε>0 andn>n 0(ε,G) everyk-graphH withn vertices and more than (π+ε) (\(\left( {\pi + \varepsilon } \right)\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)\)) edges contains a copy ofG. The real numberϱ (G) is defined in the same way adding the constraint that all independent sets of vertices inH have sizeo(n). Answering a problem of Erdős and Sós it is shown that there exist infinitely manyk-graphs with 0<ϱ(G)<π(G) for everyk≧3. It is worth noting that we were unable to find a singleG with the above property.
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This paper was written while the authors were visiting AT&T Bell Laboratories, Murray Hill, NJ 07974.