Erdős--Ko--Rado theorems in ${\ell }_2$-norm for three finite spaces

Abstract

Let $\mathcal{F}$ be a $k$-uniform hypergraph. The famous Erdős--Ko--Rado (1961) theorem determines the maximum size and extremal structure for $\mathcal{F}$ being $t$-intersecting, that is, $|F_1\cap F_2|\ge t$ for any two edges $F_1,F_2$ of $\mathcal{F}$. The codegree squared sum ${\mathrm{co}}_2(\mathcal{F})$ is the square of the ${\ell }_2$-norm of the codegree vector of all $(k-1)$-sets in $\mathcal{F}$, which was initially introduced for Turán problems of hypergraphs. Recently, Brooks and Linz (2026), as well as Wu and Zhang (2026) investigated the maximum value of ${\mathrm{co}}_2(\mathcal{F})$ and corresponding extremal structures for $\mathcal{F}$ being $t$-intersecting. Moreover, Brooks and Linz asked if the classical results on intersecting families can be extended to ${\mathrm{co}}_2(\mathcal{F})$. In this paper, by developing the spectral techniques for incidence matrices, we study the extremal problems of ${\mathrm{co}}_2(\mathcal{F})$ for $\mathcal{F}$ being intersecting families in finite vector spaces, affine spaces, and attenuated spaces, and establish the Erdős--Ko--Rado theorems in ${\ell }_2$-norm for the three finite spaces.