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 subhypergraphs have density at least d contains H. In particular, they raise the questions of determining the uniform Turán densities of 3-uniform K4- and K4. The former question was solved only recently, while the latter still remains open for almost 40 years.
In addition to K4-, the only 3-uniform hypergraphs whose uniform Turán density is known are those with zero uniform Turán density classified by Reiher, Rödl and Schacht and a specific family with uniform Turán density equal
to 1/27.
In this talk, we give an introduction to the concept of uniform Turán densities, present a way to obtain lower bounds using color schemes, and give a glimpse of the proof for determining the uniform Turán density of the tight 3-uniform cycle C_l, l ≥ 5.

This is joint work with Matija Bucić, Jacob W. Cooper, Daniel Kráľ, and David Munhá Correia.