Positive families and Boolean chains of copies of ultrahomogeneous structures

Abstract

A family of infinite subsets of a countable set X is called positive iff it is closed under supersets and finite changes and contains a co-infinite set. We show that a countable ultrahomogeneous relational structure X has the strong amalgamation property iff the setP(X)={A⊂X :A∼=X} contains a positive family. In that case the family of large copies of X (i.e. copies having infinite intersection with each orbit) is the largest positive family in P(X), and for each R-embeddable Boolean linear order Lwhose minimum is non-isolated there is a maximal chain isomorphic to L\ {minL} in 〈P(X),⊂〉. 2020 Mathematics Subject Classification. 03C15, 03C50, 20M20, 06A06, 06A05. Funding. The authors acknowledge financial support of the Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant No. 451-03-68/2020-14/200125). Manuscript received 6th April 2019, revised 27th May 2020, accepted 2nd June 2020.