[2310.09379] Some exact and asymptotic results for hypergraph Turán problems in $\ell_2$-norm Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2310.09379 (math) [Submitted on 13 Oct 2023 ( v1 ), last revised 30 Apr 2026 (this version, v4)] Title: Some exact and asymptotic results for hypergraph Turán problems in $\ell_2$-norm Authors: George Brooks , William Linz View a PDF of the paper titled Some exact and asymptotic results for hypergraph Tur\'an problems in $\ell_2$-norm, by George Brooks and 1 other authors View PDF HTML (experimental) Abstract: For a $k$-uniform hypergraph $\mathcal{H}$, the \emph{codegree squared sum} $\text{co}_2(\mathcal{H})$ is the square of the $\ell_2$-norm of the codegree vector of $\mathcal{H}$, and for a family $\mathscr{F}$ of $k$-uniform hypergraphs, the codegree squared extremal number $\text{exco}_2(n, \mathscr{F})$ is the maximum codegree squared sum of a hypergraph on $n$ vertices which does not contain any hypergraph in $\mathscr{F}$. Balogh, Clemen and Lidický recently introduced the codegree squared extremal number and determined it for a number of $3$-uniform hypergraphs, including the complete graphs $K_4^3$ and $K_5^3$. In this paper, we give a number of exact or asymptotic results for hypergraph Turán problems in the $\ell_2$-norm, including the first exact results for arbitrary $k$. Namely, we prove a version of the classical Erdős-Ko-Rado theorem for the codegree squared extremal number: if $\mathcal{F} \subset \binom{[n]}{k}$ is intersecting and $n\ge 2k$, then \[\text{co}_2(\mathcal{F}) \le \binom{n-1}{k-1}(1+(n-k+1)(k-1)),\] with equality only for the star for $n > 2k$. Our main tool is an inequality of Bey, which also gives a general upper bound on $\text{exco}_2(n, \mathscr{F})$. We also prove versions of the Erdős Matching Conjecture and the $t$-intersecting Erdős-Ko-Rado theorem for the codegree squared extremal number for large $n$, determine the exact codegree squared extremal number of minimal and linear $3$-paths and $3$-cycles, and determine asymptotically the codegree squared extremal number of minimal and linear $s$-paths and $s$-cycles for $s\ge 4$. Lastly, we derive a number of exact or asymptotic results for graph Turán-type problems in the $\ell_2$-norm from spectral extremal results for certain forbidden subgraph problems and the well-known Hofmeister's inequality. Comments: Minor revisions; to appear in European J. Combin Subjects: Combinatorics (math.CO) Cite as: arXiv:2310.09379 [math.CO] (or arXiv:2310.09379v4 [math.CO] for this version) https://doi.org/10.48550/arXiv.2310.09379 Focus to learn more arXiv-issued DOI via DataCite Submission history From: William Linz [ view email ] [v1] Fri, 13 Oct 2023 19:57:51 UTC (9 KB) [v2] Wed, 17 Jan 2024 15:03:06 UTC (14 KB) [v3] Sun, 8 Sep 2024 14:39:54 UTC (18 KB) [v4] Thu, 30 Apr 2026 12:29:42 UTC (17 KB) Full-text links: Access Paper: View a PDF of the paper titled Some exact and asymptotic results for hypergraph Tur\'an problems in $\ell_2$-norm, by George Brooks and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2023-10 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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