Abstract
In this paper we introduce a combinatorial measure of largeness for finite sets of integers (called arboricity) which is a finitary analogue of König’s Infinity Lemma. The existence of arboreal sets is shown to be very closely correlated with the Paris-Harrington version of Ramsey’s Theorem and is therefore unprovable in Peano arithmetic. Arboricity can also be exactly characterized in terms of ordinal numbers less than ε0. The ordinal characterization turns out to be only a very slight modification of the notion "α-large" introduced by Ketonen. Thus by "transitivity" this gives a simplified proof of bounds for the Ramsey-Paris-Harrington numbers previously derived by Ketonen and Solovay. Applications of arboreal sets to a selection of other combinatorial problems are also given.
Keywords
- Ordinal Number
- Winning Strategy
- Combinatorial Statement
- Peano Arithmetic
- Ramsey Number
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Research supported by NSF grant MCS-7905802
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© 1980 Springer-Verlag
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Mills, G. (1980). A tree analysis of unprovable combinatorial statements. In: Pacholski, L., Wierzejewski, J., Wilkie, A.J. (eds) Model Theory of Algebra and Arithmetic. Lecture Notes in Mathematics, vol 834. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090170
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DOI: https://doi.org/10.1007/BFb0090170
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