2023/05/19 by Hao Li, Hao Lin, Li, Hao +5 · 2 citations
Neuroscience · Mathematics · #Nuclear Receptors and Signaling #Limits and Structures in Graph Theory #Advanced Topology and Set Theory
paper · pdf · doi:10.48550/arxiv.2305.11749
The uniform Turán density π1(F) of a 3-uniform hypergraph F is the supremum over all d for which there is an F-free hypergraph with the property that every linearly sized subhypergraph with density at least d. Determining π1(F) for given hypergraphs F was suggested by Erdős and Sós in 1980s. In particular, they raised the questions of determining π1(K4(3)-) and π1(K4(3)). The former question was solved recently in [Israel J. Math. 211 (2016), 349-366] and [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter is still a major open problem. In addition to K4(3)-, there are very few hypergraphs whose uniform Turán density has been determined. In this paper, we give a sufficient condition for 3-uniform hypergraphs F satisfying π1(F)=1/4. In particular, currently all known 3-uniform hypergraphs whose uniform Turán density is 1/4, such as K4(3)- and the 3-uniform hypergraphs F⋆5 studied in [arXiv:2211.12747], satisfy this condition. Moreover, we find some intriguing 3-uniform hypergraphs whose uniform Turán density is also 1/4.