On Codegree Turán Density of the 3-Uniform Tight Cycle

C_{11}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C_{11}$$\end{document}

Abstract

Piga, Sanhueza-Matamala, and Schacht recently established that the codegree Turán density of 3-uniform tight cycles

C_{\ell }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C_\ell $$\end{document}

is 1/3 for

\ell \in \{10,13,16\}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\ell \in \{10, 13, 16\}$$\end{document}

and for all

\ell \ge 19

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\ell \ge 19$$\end{document}

. In this note, we extend their proof to determine the codegree Turán density of the 3-uniform tight cycle

C_{11}

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, thereby completing the picture for tight cycles of length at least 10.

Keywords

Turan density / Turan number / Codegree / Tight cycle / 3-uniform hypergraph / 05C65 / 05C35 / 5C38 / 05D05

References

[1] Balogh, J., Clemen, F.C., Lidický, B.: Hypergraph Turán problems in ℓ2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\ell _2$$\end{document}-norm, Surveys in Combinatorics, pp. 21–63 (2022)

[2] Czygrinowand A, Nagle B. A note on codegree problems for hypergraphs. Bull. Inst. Combin. Appl., 2001, 32: 63-69

[3] Falgas-Ravry V, Pikhurko O, Vaughan E, Volec J. The codegree threshold of K4-\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_4^-$$\end{document}. J. Lond. Math. Soc., 2023, 107: 1660-1691

[4] Mubayi D, Zhao Y. Co-degree density of hypergraphs. J. Combin. Theory Ser. A, 2007, 114: 1118-1132

[5] Piga S, Sanhueza-Matamala N, Schacht M. The codegree Turán density of 3-uniform tight cycles. J. Combin. Theory Ser. B, 2026, 176: 1-6

[6] Reiher C. Extremal problems in uniformly dense hypergraphs. Eur. J. Combin., 2020, 88 103117

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