In-Person Talk 

In the early 1980s, Erdos and Sos initiated the study of the classical Turan problem with a uniformity condition: the uniform Turan 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 Turan 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 Turan density is known are those with zero uniform Turan density classified by Reiher, Rodl and Schacht [J. London Math. Soc. 97 (2018), 77--97], and a specific family with uniform Turan density equal to 1/27.

In this talk, we give an introduction to the concept of uniform Turan densities, present a way to obtain lower bounds using color schemes, and give a glimpse of the proof for determining the uniform Turan density of the tight 3-uniform cycle C_l^{(3)}, l\ge 5.