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 only recently in [Israel J. Math. 211 (2016), 349–366] and

[J. Eur. Math. Soc. 97 (2018), 77–97], while the former still remains open for almost 40 years.

Prior to our work, the hypergraph K_4^3- was the only 3-uniform hypergraph with non-zero uniform

Turán density determined exactly. During the talk, we will present the following two results:

* We construct a family of 3-uniform hypergraphs with uniform Turán density equal to 1/27.

This answers a question of Reiher, Rödl and Schacht [J. London Math. Soc. 97 (2018), 77–97]

on the existence of such hypergraphs, stemming from their result that the uniform Turán

density of every 3-uniform hypergraph is either 0 or at least 1/27.

* We show that the uniform Turán density of the tight 3-uniform cycle of length k>=5

is equal to 4/27 if k is not divisible by three, and equal to zero otherwise.

The case k=5 resolves a problem suggested by Reiher [European J. Combin. 88 (2020), 103117].

The talk is based on results obtained jointly with (subsets of) Matija Bucić, Jacob W. Cooper,

Frederik Garbe, Ander Lamaison, Samuel Mohr and David Munhá Correia.