Minimizing the regularity of maximal regular antichains of 2- and 3-sets

Let n ≥ 3 be a natural number. We study the problem of finding the smallest r such that there is a family Α of 2-subsets and 3-subsets of [n] = {1, 2,...,n} with the following properties: (1) Α is an antichain, i.e., no member of Α is a subset of any other member of A, (2) A is maximal, i.e., for every X ∈ 2^[n]\A there is an A ∈ A with X ⊆ A or A ⊆ X, and (3) A is r-regular, i.e., every point x ∈ [n] is contained in exactly r members of A. We prove lower bounds on r, and we describe constructions for regular maximal antichains with small regularity.