On the Frankl--Tokushige conjecture and almost complete $r$-cross $t$-intersection theorems for vector spaces
Let $r\geq3$ and $k_1\geq k_2\geq\cdots\geq k_r\geq t$. Let $\mathcal{F}_1,\mathcal{F}_2,\ldots,\mathcal{F}_r$ be families of subspaces, of respective dimensions $k_1,k_2,\ldots,k_r$, in an $n$-dimensional vector space over the finite field $\mathbb{F}_q$. The $r$ families are called $r$-cross $t$-intersecting if $\dim \left(F_{1} \cap F_{2} \cap \cdots \cap F_{r}\right) \geq t$ for all $F_{i} \in \mathcal{F}_{i}, i = 1,2,\dots,r$. In 2016, Frankl and Tokushige conjectured that $\prod_{i=1}^{r}|\mathcal{F}_i|\leq\prod_{i=1}^{r}{n-1\brack k_i-1}$ for $t=1$ and $n\geq rk_1/(r-1)$. The appealing conjecture suggests establishing intersection theorems for $n\sim ck_1$ with $c=c(r)\in(1,2)$, a direction that has long been challenging. In this paper, we overcome this barrier by proving that $$\prod_{i=1}^{r}|\mathcal{F}_i|\leq\prod_{i=1}^{r}{n-t\brack k_i-t}\;\;\mbox{for all}\;\;t\geq1\;\mbox{and}\;n\geq rk_1/(r-1)+C(t,r),$$ where $C(t,r)=rt/(r-1)+1$. This proves the Frankl--Tokushige conjecture except for at most three values of $n$, and establishes an Erd\H{o}s--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.