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