Strongly maximal antichains in posets

Given a collection SS of sets, a set S∈SS∈S is said to be strongly maximal   in SS if |T∖S|≤|S∖T||T∖S|≤|S∖T| for every T∈ST∈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,…Ai,i=1,2,…, such that |Ai|=i|Ai|=i and x