Mathematical Logic Quarterly 48 (4):525-532 (2002)

Abstract
A partially ordered set is called k-homogeneous if any isomorphism between k-element subsets extends to an automorphism of . Assuming the set-theoretic assumption ⋄, it is shown that for each k, there exist partially ordered sets of size ϰ1 which embed each countable partial order and are k-homogeneous, but not -homogeneous. This is impossible in the countable case for k ≥ 4
Keywords homogeneous partial order  diamond
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DOI 10.1002/1521-3870(200211)48:4
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