Uniform Turán density of hypergraphs

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In the early 1980s, Erdős and Sós initiated the study of Turán problems with an additional uniformity condition: the uniform Turán density of a hypergraph H is the infimum over all d such for which any sufficiently large hypergraph with the property that all its linear-size subhypergraphs have density at least d contains H. In particular, Erdős and Sós raised the questions on determining the uniform Turán densities of the complete 4-vertex 3-uniform hypergraph and the complete 4-vertex 3-uniform hypergraph with an edge missing. The former remains open after almost 40 years since its statement while the latter was resolved about a decade ago only. We will survey recent progress in this area particularly focusing on corollaries of the major result of Lamaison who showed that the palette lower bound constructions are always optimal.

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