Some exact and asymptotic results for hypergraph Turán problems in ${\ell }_2$-norm

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 $\mathcal{F}$ of $k$-uniform hypergraphs, the codegree squared extremal number ${\text{exco}}_2(n,\mathcal{F})$ is the maximum codegree squared sum of a hypergraph on $n$ vertices which does not contain any hypergraph in $\mathcal{F}$. 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 $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ 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 ${\text{exco}}_2(n,\mathcal{F})$. 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.

Minor revisions; to appear in European J. Combin