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arXiv · 2607.23663

Erd\H{o}s--Ko--Rado theorems in $\ell_2$-norm for three finite spaces

Qian Bao·Yaojun Chen·Yanbo Zhang
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Abstract

Let $\mathcal{F}$ be a $k$-uniform hypergraph. The famous Erd\H{o}s--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 Tur\'an 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 Erd\H{o}s--Ko--Rado theorems in $\ell_2$-norm for the three finite spaces.

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Qian Bao, Yaojun Chen, Yanbo Zhang. 2026-07-26. Erd\H{o}s--Ko--Rado theorems in $\ell_2$-norm for three finite spaces. https://arxiv.org/abs/2607.23663

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