AN ERDOS-KO-RADO THEOREM FOR MINIMAL COVERS

作者：Ku Cheng Yeaw^*; Wong Kok Bin

来源：Bulletin of the Korean Mathematical Society, 2017, 54(3): 875-894.

DOI：10.4134/BKMS.b160334

• t-intersecting family
• Erdos-Ko-Rado
• set partitions

Let [n] = {1, 2,, n}. A set A = {A(1), A(2),, A(l)} is a minimal cover of [n] if boolean OR(1 <= i <= l) A(i) = [n] and
boolean OR(1 <= i <= l,i not equal j0) A(i) not equal [n] for all j(0) is an element of[l].
Let C(n) denote the collection of all minimal covers of [n], and write C-n = |C(n)|. Let A is an element of C(n). An element u is an element of [n] is critical in A if it appears exactly once in A. Two minimal covers A, B is an element of C(n) are said to be restricted t-intersecting if they share at least t sets each containing an element which is critical in both A and B.
A family A subset of C(n) is said to be restricted t-intersecting if every pair of distinct elements in A are restricted t-intersecting. In this paper, we prove that there exists a constant n(0) = n(0)(t) depending on t, such that for all n >= n(0), if A subset of C(n) is restricted t-intersecting, then |A| <= Cn-t. Moreover, the bound is attained if and only if A is isomorphic to the family D-0(t) consisting of all minimal covers which contain the singleton parts {1}, {t}. A similar result also holds for restricted r-cross intersecting families of minimal covers. An erdos-ko-rado theorem for minimal covers.

• 出版日期2017