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Families A 1 , ..., A k of sets are said to be cross-intersecting if A i ∩ A j = ∅ for any A i ∈ A i and A j ∈ A j , i = j. A nice result of Hilton that generalises the Erd®s-Ko-Rado (EKR) Theorem says that if r ≤ n/2 and A 1 , ..., A k are cross-intersecting sub-families of

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. This paper features two theorems. The rst one is as follows: For any two integers s and t with t ≤ s, there exists an integer k 0 (s, t) such that, for… (More)

A family H of sets is said to be hereditary if all subsets of any set in H are in H; in other words, H is hereditary if it is a union of power sets. A family A is said to be intersecting if no two sets in A are disjoint. A star is a family whose sets contain at least one common element. An outstanding open conjecture due to Chvátal claims that among the… (More)

Let G = (V, E) be a graph. For r ≥ 1, let I

For positive integers r and n with r ≤ n, let P r,n be the family of all sets of sets are said to be cross-intersecting if, for any distinct i and j in [k], any set in A i intersects any set in A j. For any r, n and k ≥ 2, we determine the cases in which the sum of sizes of cross-intersecting sub-families A 1 , A 2 , ..., A k of P r,n is a maximum, hence… (More)

A k-signed r-set on [n] = {1, ..., n} is an ordered pair (A, f), where A is an r-subset of [n] and f is a function from A to [k]. Families A 1 , ..., A p are said to be cross-intersecting if any set in any family A i intersects any set in any other family A j. Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of… (More)

We say that a set A t-intersects a set B if A and B have at least t common elements. A family A of sets is said to be t-intersecting if each set in A t-intersects any other set in A. Families A 1 , A 2 , ..., A k are said to be cross-t-intersecting if for any i and j in {1, 2, ..., k} with i = j, any set in A i t-intersects any set in A j. We prove that for… (More)

For positive integers r and n with r ≤ n, let P n,r be the family of all sets A k of sets are said to be cross-intersecting if, for any distinct i and j in [k], any set in A i intersects any set in A j. A sharp bound for the sum of sizes of cross-intersecting sub-families of P n,n has recently been established by the author. We generalise this bound by… (More)