In the early 1980s, Erdos and Sos 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 $K_4^{(3)-}$ and $K_4^{(3)}$. 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 $K_4^{(3)-}$, the only $3$-uniform hypergraphs whose uniform Turán density is known are those with zero uniform Turán density classified by Reiher, Rodl 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 $C_\ell^{(3)}$, $\ell\ge 5$.
To receive the Zoom link, please contact the seminar organiser: Maryam Sharifzadeh
