Intersecting integer partitions

If a(1), a(2),..., a(k) and n are positive integers such that n = a(1)+a(2)+ ... + a(k), then the sum a(1), a(2),..., a(k) is said to be a partition of n of length k, and a(1), a(2),..., a(k) are said to be the parts of the partition. Two partitions that differ only in the order of their parts are considered to be the same partition. Let P-n be the set of partitions of n, and let P-n,P- k be the set of partitions of n of length k. We say that two partitions t-intersect if they have at least t common parts (not necessarily distinct). We call a set A of partitions t-intersecting if every two partitions in A t-intersect. For a set A of partitions, let A(t) be the set of partitions in A that have at least t parts equal to 1. We conjecture that for n >= t, P-n(t) is a largest t-intersecting subset of P-n. We show that for k > t, P-n,P- k(t) is a largest t-intersecting subset of P-n,P- k if n <= 2k - t + 1 or n >= 3tk(5). We also demonstrate that for every t >= 1, there exist n and k such that t < k < n and P-n,P- k(t) is not a largest t-intersecting subset of P-n,P- k.