The t-intersection Problem in the Truncated Boolean Lattice

, I≤k (n, t) = I (n, t) ∩ 2 ( [n] ≤k ) . After the maximal families in I (n, t) [13] and in Ik (n, t) [1, 9] are known we study now maximal families in I≤k (n, t). We present a conjecture about the maximal cardinalities and prove it in several cases. More generally cardinalities are replaced by weights and asymptotic estimates are given. Analogous investigations are made for I (n, t) ∩ C(n, s), where C(n, s) is the class of all s-cointersecting families of subsets of [n]. In particular we establish an asymptotic form of a conjecture by Bang et al. [4].