Some exact and asymptotic results for hypergraph Turán problems in $\ell_2$-norm
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2023
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| author | Brooks, George Linz, William |
| author_facet | Brooks, George Linz, William |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_09379 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Some exact and asymptotic results for hypergraph Turán problems in $\ell_2$-norm Brooks, George Linz, William Combinatorics 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. |
| title | Some exact and asymptotic results for hypergraph Turán problems in $\ell_2$-norm |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2310.09379 |