An Erd\H os--Ko--Rado theorem for cross $t$-intersecting families

Two families $\mathcal{A}$ and $\mathcal{B}$, of $k$-subsets of an $n$-set, are {\em cross $t$-intersecting} if for every choice of subsets $A \in $\mathcal{A}$$and$B \in $\mathcal{B}$$wehave$|A \cap B| \geq t$. We address the following conjectured cross $t$-intersecting version of the Erd\H os–Ko–Rado Theorem: For all $n\ge (t+1)(k-t+1)$ the maximum value of $\mathrm{\mid }\mathcal{A}\mathrm{\mid }\mathrm{\mid }\mathcal{B}\mathrm{\mid }$ for two cross $t$-intersecting families $\mathcal{A},\mathcal{B}\subset \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ is ${\left(\genfrac{}{}{0px}{}{n-t}{k-t}\right)}^2$. We verify this for all $t\ge 14$ except finitely many $n$ and $k$ for each fixed $t$. Further, we prove uniqueness and stability results in these cases, showing, for instance, that the families reaching this bound are unique up to isomorphism. We also consider a {\em $p$-weight} version of the problem, which comes from the product measure on the power set of an $n$-set.