Palettes determine uniform Turán density

Abstract

Turán problems, which concern the minimum density threshold required for the existence of a particular substructure, are among the most fundamental problems in extremal combinatorics. We study Turán problems for hypergraphs with an additional uniformity condition on the edge distribution. This kind of Turán problems was introduced by Erdős and Sós in the 1980s but it took more than 30 years until the first non-trivial exact results were obtained when Glebov, Král' and Volec [Israel J. Math. 211 (2016), 349--366] and Reiher, Rödl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139--1159] determined the uniform Turán density of $K_4^{(3)-}$. Subsequent results exploited the powerful hypergraph regularity method, developed by Gowers and by Nagle, Rödl and Schacht about two decades ago. Central to the study of the uniform Turán density of hypergraphs are palette constructions, which were implicitly introduced by Rödl in the 1980s. We prove that palette constructions always yield tight lower bounds, unconditionally confirming present empirical evidence. This results in new and simpler approaches to determining uniform Turán densities, which completely bypass the use of the hypergraph regularity method.