Hypergraphs with vanishing Turan density in uniformly dense hypergraphs

Erdos [Israel J. Math.2 (1964) 183-190] characterised those hypergraphs F that have to appear in any sufficiently large hypergraph H of positive density. We study related questions for 3-uniform hypergraphs with the additional assumption that H has to be uniformly dense with respect to vertex sets. In particular, we characterise those hypergraphs F that are guaranteed to appear in large uniformly dense hypergraphsH of positive density. We also review the case when the density of the induced subhypergraphs of H may depend on the proportion of the considered vertex sets.