A set At-intersects a set B if A and B have at least t common
elements. A set of sets is called a family. Two families A and
B are cross-t-intersecting if each set in At-intersects each set in B. A family H is hereditary
if for each set A in H, all the subsets of A are in
H. The rth level of H, denoted by
H(r), is the family of r-element sets in H. A set
B in H is a base of H if for each set A in
H, B is not a proper subset of A. Let μ(H) denote
the size of a smallest base of H. We show that for any integers
t, r, and s with 1≤t≤r≤s, there exists an integer
c(r,s,t) such that the following holds for any hereditary family
H with μ(H)≥c(r,s,t). If A is a
non-empty subfamily of H(r), B is a non-empty
subfamily of H(s), A and B are
cross-t-intersecting, and ∣A∣+∣B∣ is maximum under
the given conditions, then for some set I in H with t≤∣I∣≤r, either A={A∈H(r):I⊆A} and B={B∈H(s):∣B∩I∣≥t},
or r=s, t<∣I∣, A={A∈H(r):∣A∩I∣≥t}, and B={B∈H(s):I⊆B}. This was conjectured by the author for t=1 and generalizes well-known
results for the case where H is a power set.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1805.0524