The Erdős-Ko-Rado Theorem in ${\mathrm{\ell }}_2$-Norm

The codegree squared sum ${\mathrm{c}\mathrm{o}}_2(\mathcal{F})$ of a family (hypergraph) $\mathcal{F}\subseteq \left(\genfrac{}{}{0px}{}{[\mathcal{n}]}{\mathcal{k}}\right)$ is defined to be the sum of codegrees squared $d(E{)}^2$ over all $E\in \left(\genfrac{}{}{0px}{}{[n]}{k-1}\right)$, where $d(E)=\mathrm{\mid }\{F\in \mathcal{F}:\mathcal{E}\subseteq \mathcal{F}\}\mathcal{\mid }$. Given a family of $k$-uniform families $\mathcal{H}$, Balogh, Clemen and Lidický recently introduced the problem to determine the maximum codegree squared sum ${\mathrm{c}\mathrm{o}}_2(\mathcal{F})$ over all $\mathcal{H}$-free $\mathcal{F}$. In the present paper, we consider the families which has as forbidden configurations all pairs of sets with intersection sizes less than $t$, that is, the well-known $t$-intersecting families. We prove the following Erdős-Ko-Rado Theorem in ${\mathrm{\ell }}_2$-norm, which confirms a conjecture of Brooks and Linz. Let $t,k,n$ be positive integers such that $t\le k\le n$. If a family $\mathcal{F}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ is $t$-intersecting, then for $n\ge (t+1)(k-t+1)$, we have $${\mathrm{c}\mathrm{o}}_2(\mathcal{F})\le \left(\genfrac{}{}{0px}{}{\mathcal{n}-\mathcal{t}}{\mathcal{k}-\mathcal{t}}\right)(\mathcal{t}+(\mathcal{n}-\mathcal{k}+\mathcal{1})(\mathcal{k}-\mathcal{t})),$$ equality holds if and only if $\mathcal{F}=\{F\in \left(\genfrac{}{}{0px}{}{[n]}{k}\right):T\subset F\}$ for some $t$-subset $T$ of $[n]$. In addition, we prove a Frankl-Hilton-Milner Theorem in ${\mathrm{\ell }}_2$-norm for $t\ge 2$, and a generalized Turán result, i.e., we determine the maximum number of copies of tight path of length 2 in $t$-intersecting families.