Uniform Turán density -- palette classification

[2505.17325] Uniform Turán density -- palette classification Skip to main content Search Submit Donate Log in Mathematics > Combinatorics arXiv:2505.17325 (math) [Submitted on 22 May 2025 ( v1 ), last revised 26 Aug 2026 (this version, v3)] Title: Uniform Turán density -- palette classification Authors: Daniel Král' , Filip Kučerák , Ander Lamaison , Gábor Tardos View PDF HTML (experimental) Abstract: 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 $K_4^{(3)-}$ and $K_4^{(3)}$. 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. Subjects: Combinatorics (math.CO) Cite as: arXiv:2505.17325 [math.CO] (or arXiv:2505.17325v3 [math.CO] for this version) https://doi.org/10.48550/arXiv.2505.17325 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Filip Kučerák [ view email ] [v1] Thu, 22 May 2025 22:37:36 UTC (84 KB) [v2] Mon, 26 May 2025 07:18:51 UTC (84 KB) [v3] Wed, 26 Aug 2026 21:35:21 UTC (84 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2025-05 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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