Combinatorics Seminar

In the early 1980s, Erdős and Sós, initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph H is the infimum over all d for which any sufficiently large hypergraph with the property that all its linear-size subhyperghraphs have density at least d contains H. In particular, they raise the questions of determining the uniform Turán densities of K_4^3, the complete 4-vertex 3-uniform hypergraph, and K_4^3-, the hypergraph K_4^3 with an edge removed. The latter question was solved in [Israel J. Math. 211 (2016), 349–366] and [J. Eur. Math. Soc. 97 (2018), 77–97], while the former still remains open for around 40 years.

In this talk, we survey recent and some very recent results concerning the uniform Turán density of hypergraphs, in particular, we present constructions of 3-uniform hypergraphs with uniform Turán density equal to 1/27, 4/27, 1/4 and 8/27, which are all non-zero values for which a construction of a 3-uniform hypergraph is known.

The talk is based on results obtained jointly with (subsets of) Matija Bucić, Jacob W. Cooper, Frederik Garbe, Daniel Iľkovič, Filip Kučerák, Ander Lamaison, Samuel Mohr and David Munhá Correia.