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

Abstract

Let $[n]$ (resp. $V$) be an $n$-element set (resp. $n$-dimensional vector space over the finite field ${\mathbb{F}}_q$), and $\left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ (resp. $\left[\genfrac{}{}{0ex}{}{V}{k}\right]$) denote the set of all $k$-subsets of $[n]$ (resp. $k$-dimensional subspaces of $V$). We say that $\mathcal{F}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ (resp. $\mathcal{F}\subseteq \left[\genfrac{}{}{0ex}{}{V}{k}\right]$) and $\mathcal{G}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k^{\prime }}\right)$ (resp. $\mathcal{G}\subseteq \left[\genfrac{}{}{0ex}{}{V}{k^{\prime }}\right]$) are $\ell$-weakly cross $t$-intersecting if ${\sum }_{1\le i,j\le \ell }|F_i\cap G_j|\ge {\ell }^{2}t-\ell +1$ (resp. ${\sum }_{1\le i,j\le \ell }\dim (F_i\cap G_j)\ge {\ell }^{2}t-\ell +1$) for all distinct $F_1,\dots ,F_{\ell }\in \mathcal{F}$ and $G_1,\dots ,G_{\ell }\in \mathcal{G}$. 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 $n\ge (2k-t+1)(t+1)+(k-t+1)k^{\prime }+k+2\ell -1$. 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.