 
Summary: An Effective Proof that Open Sets are Ramsey
Jeremy Avigad
January 22, 1996
Abstract
Solovay has shown that if O is an open subset of P() with code S
and no infinite set avoids O, then there is an infinite set hyperarithmetic
in S that lands in O. We provide a direct proof of this theorem that is
easily formalizable in ATR0.
1 Introduction
A plausible generalization of Ramsey's theorem asserts that for every two
coloring of the infinite subsets of there is an infinite homogeneous set, that is,
an infinite subset of every infinite subset of which has been assigned the same
color. Unfortunately, under the axiom of choice, this generalization is false: by
transfinite recursion along a wellordering of the reals one can cook up a color
ing with no infinite homogeneous set. On the other hand, the nonconstructive
nature of this counterexample suggests that perhaps the theorem might hold
true for colorings that are "wellbehaved" or "easily definable."
To that end, we define a partition to be a subset of the power set of ,
with the understanding that the infinite subsets falling inside the partition are
colored, say, red, and those outside the partition are colored blue. If P is a
