For large n, the maximum product is attained only by a pair both equal to the same full star of sets containing a fixed t-set, and all non-trivial extremal configurations are characterized.
Borg, The maximum product of weights of cross-intersecting families, J
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$s$-almost cross-$t$-intersecting families for finite sets
For large n, the maximum product is attained only by a pair both equal to the same full star of sets containing a fixed t-set, and all non-trivial extremal configurations are characterized.