= Non-Ramsey set from transfinite selection
Using the <axiom of choice>, enumerate all infinite subsets $A_\alpha$ of the <positive integers> in order type $2^{\aleph_0}$. By <transfinite recursion>, choose two previously unused infinite subsets $X_\alpha,Y_\alpha$ of each $A_\alpha$. This is possible because each $[A_\alpha]^\omega$ has cardinality $2^{\aleph_0}$ and fewer choices have been made at stage $\alpha$. The set $\{X_\alpha\}$ is not a <Ramsey set of infinite subsets>, since every $[A_\alpha]^\omega$ meets both it and its complement.
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