= Ramsey family in the homogeneous-cone sense
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{title2=$\exists M\in[\mathbb N]^\omega\quad [M]^\omega\subseteq\mathcal A\text{ or }[M]^\omega\cap\mathcal A=\varnothing$}
A family $\mathcal A\subseteq[\mathbb N]^\omega$ is Ramsey in the homogeneous-cone sense if there is an infinite $M\subseteq\mathbb N$ such that either $[M]^\omega\subseteq\mathcal A$ or $[M]^\omega\cap\mathcal A=\varnothing$. This is the existence of one infinite homogeneous cone. Requiring such a refinement inside every infinite initial ground <set> gives the stronger definition used for a <Ramsey set of infinite subsets>; requiring a homogeneous refinement in every finite-stem <neighbourhood> gives a <completely Ramsey set>.
These quantifiers should be distinguished. For example, the existence of a homogeneous cone on the even numbers does not constrain a family's restriction to the odd numbers. If $\mathcal E$ is the <finite-symmetric-difference parity colouring> class on the odd numbers, the family consisting of $\mathcal E$ alone has a homogeneous cone disjoint from it on the even numbers, but no homogeneous cone on the odd numbers. Every <open set> in the <ordinary topology on infinite subsets> has homogeneous cones inside all infinite ground <sets>, as follows by deciding <finite stems> through fusion.
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