On cross-intersecting uniform sub-families of hereditary families

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 such that any set in A intersects any set in B, then we say that (A,B)(A,B) is a cross-intersection pair (cip). We say that a cip (A,B)(A,B) is simple if at least one of A and B contains only one set. For a family F, let μ(F)μ(F) denote the size of a smallest set in F that is not a subset of any other set in FF. For any positive integer rr, let [r]:={1,2,...,r}[r]:={1,2,...,r}, 2[r]:={A:A⊆[r]}2[r]:={A:A⊆[r]}, F(r):={F∈F:|F|=r}F(r):={F∈F:|F|=r}. We show that if a hereditary family H⊆2[n]H⊆2[n] is compressed, μ(H)≥r+sμ(H)≥r+s with r≤sr≤s, and (A,B)(A,B) is a cip with ∅≠A⊂H(r)∅≠A⊂H(r) and ∅≠B⊂H(s)∅≠B⊂H(s), then |A|+|B||A|+|B| is a maximum if (A,B)(A,B) is the simple cip ({[r]},{B∈H(s):B∩[r]≠∅})({[r]},{B∈H(s):B∩[r]≠∅}); Frankl and Tokushige proved this for H=2[n]H=2[n]. We also show that for any 2≤r≤s2≤r≤s and m≥r+sm≥r+s there exist (non-compressed) hereditary families HH with μ(H)=mμ(H)=m such that no cip (A,B)(A,B) with a maximum value of |A|+|B||A|+|B| under the condition that ∅≠A⊂H(r)∅≠A⊂H(r) and ∅≠B⊂H(s)∅≠B⊂H(s) is simple (we prove that this is not the case for r=1r=1), and we suggest two conjectures about the extremal structures in general.peer-reviewe