A family A of sets is said to be t-intersecting if any two sets in A contain at least t commonelements. A t-intersecting family is said to be trivial if there are at least t elements common toall its sets. A family His said to be hereditary if all subsets of any set in 7-t are in H. For a finite family let F~(s)be the family of s-element sets in.F,and letn(F)thesize of a smallest set in .F that is not a subset of any other set in .F. For any two integersr and t with 1 t < r, we determine an integer no(r, t) such that, for any non-empty subset S of {t, t + 1,, r} and any finite hereditary family H with μ.(H≥no(r,t),the largestt- intersecting sub-families of the union UH(s) are trivial. The special case H= 2[n] yields asEs classical theorem of Erclifis, Ko and Rado. On the basis of the complete intersection theorem ofAhlswede and Khachatrian, we conjecture that the smallest such no (r, t) is (t 1)(r - t 1) + 1,and we show that this is true if 7-t 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.
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