Strongly maximal antichains in posets

Given a collection S of sets, a set S∈S is said to be strongly maximal in S if |T?S|≤|S?T| for every T∈S. In Aharoni (1991) [3] it was shown that a poset with no infinite chain must contain a strongly maximal antichain. In this paper we show that for countable posets it suffices to demand that the poset does not contain a copy of posets of two types: a binary tree (going up or down) or a “pyramid”. The latter is a poset consisting of disjoint antichains A_i,i=1,2,…, such that |A_i|=i and x<y whenever x∈A_i,y∈A_j and j<i (a “downward” pyramid), or x<y whenever x∈A_i,y∈A_j and i<j (an “upward” pyramid).