In the paper *[Partition theorems for systems of finite subsets of integers](https://doi.org/10.1016/0012-365X(82)90041-3)*, 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](https://doi.org/10.1112/jlms/s1-25.4.249) 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proof of the main theorem is by induction on $\omega\_1$, and the authors say that the idea of doing that came from an unpublished manuscript by Ketonen, *Set theory for a small universe, I. The Paris-Harrington Axiom.* 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?


Thanks in advance.



Looking for “Set theory for a small universe” by Ketonen
A pdf version of the 27-page manuscript by Jussi Ketonen, "Set Theory for a Small Universe, I. The Paris-Harrington Axiom", is [here on Google Drive](https://drive.google.com/file/d/1xMhBMP5Ki4RDw9pIAlsUwUh9kyMUH7xf/view?usp=sharing). The date of manuscript is perhaps 1979, or 1978.


The left margin in my hard-copy is not the best: on many lines, the first letter is truncated. But it is still readable. The scanned version captures everything in the hard-copy.


From Ketonen's introduction: 



> 
> "We will give a purely combinatorial framework for dealing with set-theoretic relations of the Harrington-Paris type; the situation will turn out to be highly analogous to the theory of large cardinals. For example the notion of 'largeness' corresponds to 'Mahloness' and the Harrington-Paris axiom transforms into a statement concerning the n-subtle cardinals of Baumgartner. ... [And conversely,] the various known large cardinal axioms naturally suggest new number-theoretical statements."
> 
> 
> 


