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hereditary families Peter Borg Department of Mathemati s University of Malta Msida MSD 2080, Malta p.borg.02 antab.net Submitted: Sep 15, 2009; A epted: Apr 5, 2010; Published: Apr 19, 2010 Mathemati s Subje t Classi ation: 05D05 Abstra t A family H of sets is hereditary if any subset of any set in H is in H. If two families A and B are su h that any set in A interse ts any set in B, then we say that (A,B) is a ross-interse tion pair ( ip). We say that a ip (A,B) is simple if at least one of A and B ontains only one set. For a family F , let μ(F) denote the size of a smallest set in F that is not a subset of any other set in F . For any positive integer r, let [r] := {1, 2, ..., r}, 2[r] := {A : A ⊆ [r]}, F (r) := {F ∈ F : |F | = r}. We show that if a hereditary family H ⊆ 2[n] is ompressed, μ(H) > r + s with r 6 s, and (A,B) is a ip with ∅ 6= A ⊂ H(r) and ∅ 6= B ⊂ H(s), then |A| + |B| is a maximum if (A,B) is the simple ip ({[r]}, {B ∈ H(s) : B ∩ [r] 6= ∅}); Frankl and Tokushige proved this for H = 2[n]. We also show that for any 2 6 r 6 s and m > r + s there exist (nonompressed) hereditary families H with μ(H) = m su h that no ip (A,B) with a maximum value of |A|+ |B| under the ondition that ∅ 6 = A ⊂ H(r) and ∅ 6= B ⊂ H(s) is simple (we prove that this is not the ase for r = 1), and we suggest two onje tures about the extremal stru tures in general. 1 Introdu tion We shall use small letters su h as x to denote elements of a set or positive integers, apital letters su h as X to denote sets, and alligraphi letters su h as F to denote families (i.e. sets whose members are sets themselves). Unless otherwise stated, it is to be assumed that sets and families are nite. N is the set {1, 2, ...} of positive integers. For m,n ∈ N with m 6 n, the set {i ∈ N : m 6 i 6 n} is denoted by [m,n], and if m = 1 then we also write [n]. The power set {A : A ⊆ X} of a set X is denoted by 2 , and {A ⊆ X : |A| = r} is denoted by (X r ).