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 ^Ai,i=1,2,..., such that |^Ai|=i and x<y whenever x∈^Ai,y∈^Aj and j<i (a "downward" pyramid), or x<y whenever x∈^Ai, y∈^Aj and i<j (an "upward" pyramid).