2025/05/22 by Král', Daniel, Kučerák, Filip, Lamaison, Ander +1
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.17325
In the 1980s, Erdős and Sós initiated the study of Turán hypergraph problems with a uniformity condition on the distribution of edges, i.e., determining density thresholds for the existence of a hypergraph H in a host hypergraph with edges uniformly distributed. In particular, Erdős and Sós asked to determine the uniform Turán densities of the hypergraphs K4(3)- and K4(3). After more than 30 years, the former was solved by Glebov, Král' and Volec [Israel J. Math. 211 (2016), 349-366] and Reiher, Rödl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter still remains open. In these two cases and several additional cases, the tight lower bounds are provided by a so-called palette construction. Lamaison [arXiv:2408.09643] has recently showed that the uniform Turán density of a 3-uniform hypergraph H is equal to the supremum of the densities of palettes that H is not colorable with. We give a necessary and sufficient condition, which is easy to verify, on the existence of a 3-uniform hypergraph colorable by a set of palettes and not colorable by another given set of palettes. We also demonstrate how our result can be used to prove the existence of 3-uniform hypergraphs with specific values of the uniform Turán density.