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Two families A and B of sets are cross-t-intersecting if each set
in A intersects each set in B in at least t elements. A family H is
hereditary if for each set A in H, all the subsets of A are in H. Let
H(r) denote the family of r-element sets in 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 whose maximal sets are of size at least c(r, s, t). If A is a nonempty 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}. We give c(r, s, t) explicitly. The result was conjectured by the
author for t = 1 and generalizes well-known results for the case
where H is a power set.peer-reviewe
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