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proof of the main theorem is by induction on $\omega_1$, and the authors say that the idea of doing that came from an unpublished manuscript by Ketonen, Set theory for a small universe, I. The Paris-Harrington Axiom.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

Thanks in advance.

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

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. 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."

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    $\begingroup$ Given the popular claim for the paper, I planned to comment that if you finally did the scan, you'd be the hero of the day. Now you are, many thanks. By the way, I was neither successful looking for contact info of Ketonen. $\endgroup$ – Pedro Sánchez Terraf Nov 15 '18 at 22:58
  • $\begingroup$ 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. $\endgroup$ – Asaf Karagila Nov 16 '18 at 10:15
  • $\begingroup$ A belated realization, and full disclosure. I had totally forgotten that my name was mentioned in Ketonen & Solovay's "Rapidly Growing Ramsey Functions", Annals of Mathematics, 113(2), 1981. And that Takeuti's 1987 edition of Proof Theory refers to the "Ketonen-Solovay-Quinsey result on Paris-Harrington's theorem". Notwithstanding, I believe my contribution was peripheral. $\endgroup$ – jeq Dec 31 '18 at 6:00

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