On rr-cross tt-intersecting families for vector spaces

Let $V$ be a vector space over a finite field $\mathbb{F}_q$ with dimension $n$, and ${V\brack k}$ the set of all subspaces of $V$ with dimension $k$. The families $\mathcal{F}_1,\dots,\mathcal{F}_r\subset{V\brack k}$ are called $r$-cross $t$-intersecting families if $\dim(F_1\cap F_2\cap\dots\cap F_r)\ge t$ for any $F_i\in\mathcal{F}_i$, $i=1,2,\dots,r$. In this paper, we prove a product version of the Hilton-Milner theorem for vector spaces, determining the structure of $2$-cross $t$-intersecting families $\mathcal{F}$, $\mathcal{G}$, with the maximum product of their sizes under the condition that both $\dim(\cap\{F: F\in\mathcal{F}\})$ and $\dim(\cap\{G: G\in\mathcal{G}\})$ are less than $t$. We also characterize the structure of $r$-cross $t$-intersecting families $\mathcal{F}_1$, $\mathcal{F}_2$, $\dots$, $\mathcal{F}_r$ with the maximum product of their sizes under the condition that $r\ge3$ and $\dim(\cap\{F: F\in\mathcal{F}_i\})<t$ for any $i\in\{1,2,\dots,r\}$

The Erdős-Ko-Rado basis for a Leonard system

We introduce and discuss an Erd\H{o}s-Ko-Rado basis for the underlying vector space of a Leonard system $\Phi = (A; A^*; \{E_i\}_{i=0}^d ; \{E_i^* \}_{i=0}^d)$ that satisfies a mild condition on the eigenvalues of $A$ and $A^*$. We describe the transition matrices to/from other known bases, as well as the matrices representing $A$ and $A^*$ with respect to the new basis. We also discuss how these results can be viewed as a generalization of the linear programming method used previously in the proofs of the "Erd\H{o}s-Ko-Rado theorems" for several classical families of $Q$-polynomial distance-regular graphs, including the original 1961 theorem of Erd\H{o}s, Ko, and Rado

The maximum sum of sizes of non-empty cross tt-intersecting families

Let $[n]:=\lbrace 1,2,\ldots,n \rbrace$, and $M$ be a set of positive integers. Denote the family of all subsets of $[n]$ with sizes in $M$ by $\binom{\left[n\right]}{M}$. The non-empty families $\mathcal{A}\subseteq\binom{\left[n\right]}{R}$ and $\mathcal{B}\subseteq \binom{\left[n\right]}{S}$ are said to be cross $t$-intersecting if $|A\cap B|\geq t$ for all $A\in \mathcal{A}$ and $B\in \mathcal{B}$. In this paper, we determine the maximum sum of sizes of non-empty cross $t$-intersecting families, and characterize the extremal families. Similar result for finite vector spaces is also proved