We say that a set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. Two families $\mathcal{ A}$ and $\mathcal{ B}$ of sets are said to be cross-$t$-intersecting if each set in $\mathcal{ A}$ $t$-intersects each set in $\mathcal{ B}$. A subfamily $\mathcal{ S}$ of a family $\mathcal{ F}$ is called a $t$-star of $\mathcal{ F}$ if the sets in $\mathcal{ S}$ have $t$ common elements. Let $l(\mathcal{ F},t)$ denote the size of a largest $t$-star of $\mathcal{ F}$. We call $\mathcal{ F}$ a $(\leq r)$-family if each set in $\mathcal{ F}$ has at most $r$ elements. We determine a function $c : \mathbb{ N}^3 \rightarrow \mathbb{ N}$ such that the following holds. If $\mathcal{ A}$ is a subfamily of a $(\leq r)$-family $\mathcal{ F}$ with $l(\mathcal{ F},t) \geq c(r,s,t)l(\mathcal{ F},t+1)$, $\mathcal{ B}$ is a subfamily of a $(\leq s)$-family $\mathcal{ G}$ with $l(\mathcal{ G},t) \geq c(r,s,t)l(\mathcal{ G},t+1)$, and $\mathcal{ A}$ and $\mathcal{ B}$ are cross-$t$-intersecting, then $|\mathcal{ A}||\mathcal{ B}| \leq l(\mathcal{ F},t)l(\mathcal{ G},t)$. Some known results follow from this, and we identify several natural classes of families for which the bound is attained.
Categories: math.CO