On the Frankl--Tokushige conjecture and almost complete $r$-cross $t$-intersection theorems for vector spaces

Let $r\ge 3$ and $k_1\ge k_2\ge \cdots \ge k_r\ge t$. Let ${\mathcal{F}}_1,{\mathcal{F}}_2,\dots ,{\mathcal{F}}_r$ be families of subspaces, of respective dimensions $k_1,k_2,\dots ,k_r$, in an $n$-dimensional vector space over the finite field ${\mathbb{F}}_q$. The $r$ families are called $r$-cross $t$-intersecting if $\dim (F_1\cap F_2\cap \cdots \cap F_r)\ge t$ for all $F_i\in {\mathcal{F}}_i,i=1,2,\dots ,r$. In 2016, Frankl and Tokushige conjectured that ${\prod }_{i=1}^r\mathrm{\mid }{\mathcal{F}}_i\mathrm{\mid }\le {\prod }_{i=1}^r\left[\genfrac{}{}{0px}{}{n-1}{k_i-1}\right]$ for $t=1$ and $n\ge r{k}_1\mathrm{/}(r-1)$. The appealing conjecture suggests establishing intersection theorems for $n\sim c{k}_1$ with $c=c(r)\in (1,2)$, a direction that has long been challenging. In this paper, we overcome this barrier by proving that $$\prod _{i=1}^r\mathrm{\mid }{\mathcal{F}}_i\mathrm{\mid }\le \prod _{i=1}^r\left[\genfrac{}{}{0px}{}{n-t}{k_i-t}\right]forallt\ge 1andn\ge r{k}_1\mathrm{/}(r-1)+C(t,r),$$ where $C(t,r)=rt\mathrm{/}(r-1)+1$. This proves the Frankl–Tokushige conjecture except for at most three values of $n$, and establishes an Erdős–Ko–Rado type theorem for almost all values of parameters. Furthermore, we characterize all extremal configurations. Our proof is purely combinatorial and based on the $t$-cover method, with several essential refinements. We also obtain almost complete intersection theorems for $r$-wise $t$-intersecting families and non-trivial $r$-cross $t$-intersecting families.