Discrete Math Seminar: Uniform Turán density

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 K4(3)−K_4^{(3)-}K4(3)−​ and K4(3)K_4^{(3)}K4(3)​. The former question was solved only recently in [Israel J. Math. 211 (2016), 349–366] and [J. Eur. Math. Soc. 97 (2018), 77–97], while the latter still remains open for almost 40 years. In addition to K4(3)−K_4^{(3)-}K4(3)−​, the only $3$-uniform hypergraphs whose uniform Turán density is known are those with zero uniform Turán density classified by Reiher, Rödl and Schacht [J. London Math. Soc. 97 (2018), 77–97] 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 Cℓ(3)C_\ell^{(3)}Cℓ(3)​, $\ell \ge 5$.