It is consistent relative to large cardinals that every countable-to-one coloring c of pairs from ω₂ has a closed subset A of order type ω₁ on which c is injective.
Rainbow general- izations of Ramsey theory: a survey
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A Topological Rainbow Ramsey Theorem
It is consistent relative to large cardinals that every countable-to-one coloring c of pairs from ω₂ has a closed subset A of order type ω₁ on which c is injective.