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).