## Journal of Symbolic Logic

### Ramsey's Theorem for Computably Enumerable Colorings

#### Abstract

It is shown that for each computably enumerable set $\mathscr{P}$ of n-element subsets of $\omega$ there is an infinite $\Pi^0_n$ set $A \subseteq \omega$ such that either all n-element subsets of A are in $\mathscr{P}$ or no n-element subsets of A are in $\mathscr{P}$. An analogous result is obtained with the requirement that A be $\Pi^0_n$ replaced by the requirement that the jump of A be computable from $0^{(n)}$. These results are best possible in various senses.

#### Article information

Source
J. Symbolic Logic, Volume 66, Issue 2 (2001), 873-880.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183746479

Mathematical Reviews number (MathSciNet)
MR1833484

Zentralblatt MATH identifier
0988.03066

JSTOR