AI 中文总结
研究有限向量空间、仿射空间和衰减空间中相交族超图的共度数平方和 $\mathrm{co}_2(\mathcal{F})$ 的极值问题，通过发展关联矩阵谱技术，建立了这三个有限空间中 $\ell_2$-范数下的 Erdős--Ko--Rado 定理。
AI 中文摘要
设 $\mathcal{F}$ 为 $k$-一致超图。著名的 Erdős--Ko--Rado（1961）定理确定了 $\mathcal{F}$ 为 $t$-相交（即对任意两条边 $F_1,F_2$，有 $|F_1 \cap F_2| \ge t$）时的最大规模和极值结构。共度数平方和 $\mathrm{co}_2(\mathcal{F})$ 是 $\mathcal{F}$ 中所有 $(k - 1)$-集的共度数向量的 $\ell_2$-范数的平方，最初用于超图的 Turán 问题。近期 Brooks 和 Linz 以及 Wu 和 Zhang 研究了 $\mathcal{F}$ 为 $t$-相交时 $\mathrm{co}_2(\mathcal{F})$ 的最大值及相应极值结构，且 Brooks 和 Linz 提出能否将相交族的经典结果扩展到 $\mathrm{co}_2(\mathcal{F})$。本文通过发展关联矩阵的谱技术，研究了有限向量空间、仿射空间和衰减空间中 $\mathcal{F}$ 为相交族时 $\mathrm{co}_2(\mathcal{F})$ 的极值问题，并建立了这三个有限空间中 $\ell_2$-范数下的 Erdős--Ko--Rado 定理。
英文摘要
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.
