On $\ell$-weakly cross $t$-intersecting families for sets and vector spaces

TL;DR

This paper investigates $ ext{ell}$-weakly cross $t$-intersecting families of sets and vector spaces, providing an alternative proof of key theorems and establishing bounds on the product of their sizes.

Contribution

It offers a new proof for the set version of the $ ext{ell}$-weakly cross $t$-intersecting theorem and derives explicit bounds for the sizes of such families.

Findings

Provided an alternative proof of the set version of the $ ext{ell}$-weakly cross $t$-intersecting theorem.

Established an explicit lower bound for $n$ in the context of these families.

Proved an upper bound on the product of sizes of $ ext{ell}$-weakly cross $t$-intersecting subspace families.

Abstract

Let $[n]$ (resp. $V$) be an $n$-element set (resp. $n$-dimensional vector space over the finite field $\mathbb{F}_{q}$), and $\binom{[n]}{k}$ (resp. $\genfrac{[}{]}{0pt}{}{V}{k}$) denote the set of all $k$-subsets of $[n]$ (resp. $k$-dimensional subspaces of $V$). We say that $\mathcal{F}\subseteq\binom{[n]}{k}$ (resp. $\mathcal{F}\subseteq\genfrac{[}{]}{0pt}{}{V}{k}$) and $\mathcal{G}\subseteq\binom{[n]}{k'}$ (resp. $\mathcal{G}\subseteq \genfrac{[}{]}{0pt}{}{V}{k'}$) are $\ell$-weakly cross $t$-intersecting if $\sum_{1\le i,j\le \ell}|F_i\cap G_j|\geq \ell^{2}t-\ell+1$ (resp. $\sum_{1\le i,j\le \ell}\dim(F_i\cap G_j)\geq \ell^{2}t-\ell+1$) for all distinct $F_1,\ldots,F_{\ell}\in\mathcal{F}$ and $G_1,\ldots,G_{\ell}\in\mathcal{G}$. In this paper, we provide an alternative proof of the set version of the $\ell$-weakly cross $t$-intersecting theorem and an explicit lower bound for $n$. Moreover, we prove that if $\mathcal{F}$ and $\mathcal{G}$ are $\ell$-weakly cross $t$-intersecting subspace families, then \[ |\mathcal{F}| \cdot |\mathcal{G}| \leq\genfrac{[}{]}{0pt}{}{n-t}{k-t}\genfrac{[}{]}{0pt}{}{n-t}{k'-t} \] holds, provided that $n\geq (2k-t+1)(t+1)+(k-t+1)k'+k+2\ell-1$. This extends the theorem of Cao, Lu, Lv and Wang [J. Combin. Theory Ser. A 193 (2023), 105688], who established the upper bound for the product of the sizes of cross $t$-intersecting subspace families.