= Stem-supported Ramsey family need not be completely Ramsey
Let E be the even <positive integers>, and take a <finite-symmetric-difference parity colouring> $\varepsilon$ of $[E]^\omega$. The family $\{\{1\}\cup X:X\in[E]^\omega,\ \varepsilon(X)=0\}$ is Ramsey even under the every-ground-set convention, since every infinite ground set can be thinned to omit 1. It is not <completely Ramsey>: inside $[\{1\},E]$, every infinite tail has two subfamilies of opposite parity. Its supporting <neighbourhood> is closed and <nowhere dense> in the <ordinary topology on infinite subsets>, but open in the <Ellentuck topology>.
Back to article page
