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)  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).
- Strongly maximal
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics