The Erdős--Ko--Rado Theorem in $\ell_2$-Norm

TL;DR

This paper proves an Erdős–Ko–Rado-type theorem in the $\ell_2$-norm for $t$-intersecting $k$-uniform hypergraphs, establishing a sharp bound ${\rm co}_2(\mathcal{F}) \le {\binom{n-t}{k-t}}\bigl(t+(n-k+1)(k-t)\bigr)$ for all $n \ge (t+1)(k-t+1)$, with equality if and only if $\mathcal{F}$ is a full $t$-star. The authors adapt the generating-set method of Ahlswede–Kleitman (AK1996) to the $\ell_2$ setting and leverage left-compression to reduce to left-compressed extremals, together with Bey-type inequalities relating codegree sums to pairwise intersections. They also derive a Frankl–Hilton–Milner-type result in $\ell_2$-norm for $t\ge 2$, showing that the maximal co_2 among nontrivial $t$-intersecting families is attained by $\mathcal{H}(n,k,t)$ or $\mathcal{A}(n,k,t)$, and prove a generalized Turán-type bound counting copies of a tight path of length 2. As corollaries, the paper determines the maximal number of tight-path copies in these families and connects the extremal configurations to full $t$-stars, extending classical EKR phenomena to the $\ell_2$ framework with explicit equality cases.

Abstract

The codegree squared sum ${\rm co}_2(\cal F)$ of a family (hypergraph) $\cal F \subseteq \binom{[n]} k$ is defined to be the sum of codegrees squared $d(E)^2$ over all $E\in \binom{[n]}{k-1}$, where $d(E)=|\{F\in \cal F: E\subseteq F\}|$. Given a family of $k$-uniform families $\mathscr H$, Balogh, Clemen and Lidický recently introduced the problem to determine the maximum codegree squared sum ${\rm co}_2(\cal F)$ over all $\mathscr H$-free $\cal F$. 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 $\ell_2$-norm, which confirms a conjecture of Brooks and Linz. Let $t,k,n$ be positive integers such that $t\leq k\leq n$. If a family $\mathcal F\subseteq \binom{[n]}{k}$ is $t$-intersecting, then for $n\ge (t+1)(k-t+1)$, we have \[{\rm co}_2(\cal F)\le {\binom{n-t}{k-t}}(t+(n-k+1)(k-t)),\] equality holds if and only if $\mathcal{F}=\{F\in {\binom{[n]}{k}}: T\subset F\}$ for some $t$-subset $T$ of $[n]$. In addition, we prove a Frankl--Hilton--Milner Theorem in $\ell_2$-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.

Theorems & Definitions (31)

• Theorem 1.1: Erdős--Ko--Rado, EKR1961
• Theorem 1.2: Bey, Bey
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: Hilton--Milner, HM
• Theorem 1.6: FranklFra1978, Frankl and FürediFF, and Ahlswede and KhachatrianAK1996
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• proof