A set of sets is called a family  . Two families AA and BB are said to be cross-  tt-intersecting   if each set in AA intersects each set in BB in at least tt elements. For a family FF, let l(F,t)l(F,t) denote the size of a largest subfamily of FF whose sets have at least tt common elements. We call FF a (≤r)(≤r)-family   if each set in FF has at most rr elements. We show that for any positive integers rr, ss and tt, there exists an integer c(r,s,t)c(r,s,t) such that the following holds. If AA is a subfamily of a (≤r)(≤r)-family FF with l(F,t)≥c(r,s,t)l(F,t+1)l(F,t)≥c(r,s,t)l(F,t+1), BB is a subfamily of a (≤s)(≤s)-family GG with l(G,t)≥c(r,s,t)l(G,t+1)l(G,t)≥c(r,s,t)l(G,t+1), and AA and BB are cross-tt-intersecting, then |A||B|≤l(F,t)l(G,t)|A||B|≤l(F,t)l(G,t). We give c(r,s,t)c(r,s,t) explicitly. Some known results follow from this, and we identify several natural classes of families for which the bound is attained.
['Peter Borg']
The maximum product of sizes of cross-intersecting families