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 $B \in \mathcal{ B}$ we have $|A \cap B| \geq t$. We address the following conjectured cross $t$-intersecting version of the Erd\H os--Ko--Rado Theorem: For all $n \geq (t+1)(k-t+1)$ the maximum value of $|\mathcal{ A}||\mathcal{ B}|$ for two cross $t$-intersecting families $\mathcal{ A}, \mathcal{ B} \subset\binom{ [n]}{ k}$ is $\binom{ n-t}{ k-t}^2$. We verify this for all $t \geq 14$ 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.
Categories: math.CO