Extremal t-intersecting sub-families of hereditary families

A family A of sets is said to be t-intersecting if any two sets in A contain at least t common elements. A t-intersecting family is said to be trivial if there are at least t elements common to all its sets. A family H is said to be hereditary if all subsets of any set in H are in H. For a finite family F , let F (s) be the family of s-element sets in F , and let μ(F) be the size of a smallest set in F that is not a subset of any other set in F . For any two integers r and t with 1 t < r, we determine an integer n0(r, t) such that, for any non-empty subset S of {t, t + 1, . . . , r} and any finite hereditary family H with μ(H) n0(r, t), the largest tintersecting sub-families of the union ⋃ s∈S H(s) are trivial. The special case H = 2 yields a classical theorem of Erdős, Ko and Rado. On the basis of the complete intersection theorem of Ahlswede and Khachatrian, we conjecture that the smallest such n0(r, t) is (t + 1)(r − t + 1) + 1, and we show that this is true if H is compressed. We apply our main result to obtain new results on t-intersecting families of signed sets, permutations and separated sets. This work supports some open conjectures.