Uncountable homogeneous partial orders

A partially ordered set (P, less than or equal to) is called k-homogeneous if any isomorphism between k-element subsets extends to an automorphism of (P, less than or equal to). Assuming the set-theoretic assumption square(N-1), it is shown that for each k, there exist partially ordered sets of size N-1 which embed each countable partial order and axe k-homogeneous, but not (k + 1)-homogeneous. This is impossible in the countable case for k greater than or equal to 4.