Countable Length Ramsey Games

Abstract

It is known that the first player has a winning strategy for the Ramsey game R(α, < n, a) if α < ω² is a limit ordinal, and for the Ramsey game R(α, 2,α) if α < ω₁ is a limit ordinal or the successor of a limit ordinal. By way of contrast, we show that the second player wins R(ω ^ω , 3, ω^ω). More generally, the second player wins R(φ, 3, ω V (ω ^ω)*) for φ a scattered linear order type of any cardinality (where the game lasts ω-many moves).

This work was partially supported by NSF Grant DMS 9626713.