Antichains of Copies of Ultrahomogeneous Structures

Abstract

We investigate possible cardinalities of maximal antichains in the poset of copies P(X), ⊂ of a countable ultrahomogeneous relational structure X. It turns out that if the age of X has the strong amalgamation property, then, defining a copy of X to be large iff it has infinite intersection with each orbit of X, the structure X can be partitioned into countably many large copies, there are almost disjoint families of large copies of size continuum and, hence, there are (maximal) antichains of size continuum in the poset P(X). Finally, we show that the posets of copies of all countable ultrahomogeneous partial orders contain maximal antichains of cardinality continuum and determine which of them contain countable maximal antichains. That holds, in particular, for the random (universal ultrahomogeneous) poset.

The poset of copies for automorphism groups of countable relational structures

“…Results on the theme of copies of an homogeneous structure have been developped in a series of papers by Kurilić, Kuzeljević and Todorcević (e.g. [9,10,11,12]). Some of the results presented here have been the subject of a lecture by the third author in March 2019 at the FG1 Seminar of the Technical University of Vienna [18].…”