An Analogue of the Hilton-Milner Theorem for Weak Compositions

Abstract

Abstract. Let N 0 be the set of non-negative integers, and let P (n, l) denote the set of all weak compositions of n with l parts, i.e., P (n, l) = {(x 1 , x 2 , . . . , x l ) ∈ N l 0 :A family A ⊆ P (n, l) is said to be trivially t-intersecting if there is a t-set T of [l] = {1, 2, . . . , l} and elements ys ∈ N 0 (s ∈ T ) such that A = {u ∈ P (n, l) : u(j) = y j for all j ∈ T }. We prove that given any positive integers l, t with l ≥ 2t + 3, there exists a constant n 0 (l, t) depending only on l and t, such that for all n ≥ n 0 (l, t), if A ⊆ P (n, l) is non-trivially t-intersecting, then Moreover, equality holds if and only if there is a t-set T of [l] such thatwhere As = {u ∈ P (n, l) : u(j) = 0 for all j ∈ T and u(s) = 0}and q i ∈ P (n, l) with q i (j) = 0 for all j ∈ [l] \ {i} and q i (i) = n.

A Deza–Frankl Type Theorem for Set Partitions

“…The investigation of the Erdős-Ko-Rado property for graphs started in [23], and gave rise to [4,6,21,22,24,47]. The Erdős-Ko-Rado type results also appear in vector spaces [9,18], set partitions [27,29,30] and weak compositions [32,33,34].…”