## Abstract

Let*H*
^{r} be an*r*-uniform hypergraph. Let*g*=*g*(*n*;*H*
^{r}) be the minimal integer so that any*r*-uniform hypergraph on*n* vertices and more than*g* edges contains a subgraph isomorphic to*H*
^{r}. Let*e* =*f*(*n*;*H*
^{r},*εn*) denote the minimal integer such that every*r*-uniform hypergraph on*n* vertices with more than*e* edges and with no independent set of*εn* vertices contains a subgraph isomorphic to*H*
^{r}.

We show that if*r*>2 and*H*
^{r} is e.g. a complete graph then

while for some*H*
^{r} with\(\mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)^{ - 1} g(n;H^r ) \ne 0\)

. This is in strong contrast with the situation in case*r*=2. Some other theorems and many unsolved problems are stated.

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*Dedicated to Tibor Gallai on his seventieth birthday*