Seminar: Uniform Turán density
Date/Time: | Thursday, 28 Oct 2021 from 2:10 pm to 3:00 pm |
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Location: | 401 Carver Hall |
Cost: | Free |
URL: | https://seminar.mathematicaster.org/ |
Contact: | Bernard Lidický |
Phone: | 515-294-8136 |
Channel: | Research |
Categories: | Lectures |
Actions: | Download iCal/vCal | Email Reminder |
Join this in-person seminar in 401 Carver Hall to learn more.
Abstract: In the early 1980s, Erdos and Sós initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph HHH is the infimum over all ddd for which any sufficiently large hypergraph with the property that all its linear-size subhyperghraphs have density at least ddd contains HHH. 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 333-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 1/271/271/27. 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 333-uniform cycle Cl(3)C_ell^{(3)}Cl(3), l>=5ellge 5l>=5.