An Analogue of the Hilton-milner Theorem for Weak Compositions

Abstract. Let N0 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) = {(x1, x2, . . . , xl) ∈ N0 : x1 + x2 + · · · + xl = n}. For any element u = (u1, u2, . . . , ul) ∈ P (n, l), denote its ith-coordinate by u(i), i.e., u(i) = ui. A family A ⊆ P (n, l) is said to be t-intersecting if |{i : u(i) = v(i)}| ≥ t for all u,v ∈ A. 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 ∈ N0 (s ∈ T ) such that A = {u ∈ P (n, l) : u(j) = yj for all j ∈ T}. We prove that given any positive integers l, t with l ≥ 2t + 3, there exists a constant n0(l, t) depending only on l and t, such that for all n ≥ n0(l, t), if A ⊆ P (n, l) is non-trivially t-intersecting, then