  P. Erd\H{o}s [On extremal problems of graphs and generalized graphs, Israel
Journal of Mathematics 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 hypergraphs $H$ 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.
