The maximum product of weights of cross-intersecting families.

A set of sets is called a family. Two families A and B are said to be cross-t-intersecting if each set in A intersects each set in B in at least t elements. An active problem in extremal set theory is to determine the maximum product of sizes of cross-t-intersecting subfamilies of a given family. This incorporates the classical Erdõs-Ko-Rado (EKR) problem. We prove a cross-t-intersection theorem for weighted subsets of a set by means of a new subfamily alteration method, and use the result to provide solutions for three natural families. For r ∊ [n] = {1, 2, . . . , n}, let ([n] r) be the family of r-element subsets of [n], and let ([n] ⩽r ) be the family of subsets of [n] that have at most r elements. Let Fn,r,t be the family of sets in ([n] ⩽r ) that contain [t]. We show that if g : ([m] ⩽r ) → R+ and h : ([n] ⩽ s ) → R+ are functions that obey certain conditions, A ⊆ ([m] ⩽ r ) , B ⊆ ([n] ⩽ s ) , and A and B are cross-t-intersecting, then and equality holds if A = Fm,r,t and B = Fn,s,t. We prove this in a more general setting and characterize the cases of equality. We use the result to show that the maximum product of sizes of two cross-t-intersecting families A ⊆ ([m] r ) and B ⊆ ([n] s ) is (m-t/r-t )(n-t/s-t ) for min{m, n} ⩾ n0(r, s, t), where n0(r, s, t) is close to best possible. We obtain analogous results for families of integer sequences and for families of multisets. The results yield generalizations for k ⩾ 2 cross-t-intersecting families, and EKR-type results. [ABSTRACT FROM AUTHOR]