# Antichains of Copies of Ultrahomogeneous Structures

## TL;DR

This paper explores the sizes of maximal antichains in the poset of copies of countable ultrahomogeneous structures, revealing conditions under which large antichains of continuum size exist, especially for structures with the strong amalgamation property.

## Contribution

It characterizes the cardinalities of maximal antichains in the poset of copies of countable ultrahomogeneous structures, including the existence of continuum-sized antichains under certain conditions.

## Key findings

- Countable ultrahomogeneous structures with strong amalgamation have continuum-sized antichains.
- Posets of copies of all countable ultrahomogeneous partial orders contain continuum-sized maximal antichains.
- The random ultrahomogeneous poset has maximal antichains of continuum size and some with countable size.

## Abstract

We investigate possible cardinalities of maximal antichains in the poset of copies $\langle \mathbb P(\mathbb X),\subset \rangle$ of a countable ultrahomogeneous relational structure $\mathbb X$. It turns out that if the age of $\mathbb X$ has the strong amalgamation property, then, defining a copy of $\mathbb X$ to be large iff it has infinite intersection with each orbit of $\mathbb X$, the structure $\mathbb X$ 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 $\mathbb P (\mathbb X)$. 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.