The t-intersection Problem in the Truncated Boolean Lattice

AbstractLet I(n, t) be the class of all t -intersecting families of subsets of [ n ] and set Ik(n, t) =I (n, t) ∩ 2[ n ]k, I≤k(n, t) =I(n, t) ∩ 2[ n ] ≤k.After the maximal families inI (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]