# Looking for “Set theory for a small universe” by Ketonen

In the paper Partition theorems for systems of finite subsets of integers, Pudlák and Rödl show a Ramsey-type result. The main feature of this result is that the sizes of sets in such systems are not fixed in advance (as in Ramsey's original theorem and Erdős and Rado's generalization to arbitrary partitions of $$[\mathbb{N}]^k$$).

The proof of the main theorem is by induction on $$\omega_1$$, and the authors say that the idea of doing that came from an unpublished manuscript by Ketonen, Set theory for a small universe, I. The Paris-Harrington Axiom. Does anyone here has access to this manuscript?

• @PedroSánchezTerraf The paper might be close to the paper Rapidly growing Ramsey functions'' by Ketonen-Solovay. – Mohammad Golshani Nov 15 '18 at 3:49
• The Paris-Harrington result was published in 1977, the Pudlak-Rold paper in 1982, so apparently this paper can be dated in between. – Matt F. Nov 15 '18 at 3:58
• Is it possible to contact Ketonen to ask if the manuscript can be posted more publicly? – David Roberts Nov 15 '18 at 6:18
• @MohammadGolshani Thank you very much for the new reference. DavidRoberts: I'll try to contact Ketonen, that's always a good idea. – Pedro Sánchez Terraf Nov 15 '18 at 10:44
• @DavidRoberts As I commented below, I was neither successful looking for contact info of Ketonen. – Pedro Sánchez Terraf Nov 15 '18 at 23:01

• Those pages are not jam-packed. I think it should be fairly straightforward typing this into $\rm\LaTeX$ by someone who is willing to spend a few hours. – Asaf Karagila Nov 16 '18 at 10:15