## Combinatorial unprovability proofs and their model-theoretic counterparts.(English)Zbl 1301.03062

The paper begins with a comprehensive survey of unprovability in PA of Ramsey-type combinatorial principles, and then proceeds with a model-theoretic analysis of the canonical Ramsey theorem with a largeness condition ERL, and of a version the Kanamori-McAloon principle KM. The ERL principle states that for all $$n$$ and $$k$$ there is an $$m$$ such that for every function $$f$$ defined on $$[m]^n$$ there is an $$H\subseteq m$$, with $$|H|>\max\{\min(H),k\}$$, on which $$f$$ is canonical, i.e.  there is a $$v\subseteq n$$ such that for all $$x_1<\cdots < x_n$$ in $$H$$, $$f(x_1,\dots, x_n)$$ depends only on $$x_i$$ for $$i\in v$$. The KM$$_j$$ principle is: For all $$n$$ and $$k$$ there is an $$m$$ such that whenever $$f:[m]^n\rightarrow{\mathbb N}$$ is $$j$$-regressive, i.e.  for all $$x_1<\cdots <x_n\leq m$$, $$f(x_1,\dots, x_n)\in [x_{j-1}, x_j]$$, then there is an $$H\subseteq m$$ such that $$|H|\geq k$$ and the values of $$f$$ on $$[H]^n$$ depend only on $$x_1,\dots, x_j$$.
Carlucci and Weiermann gave a combinatorial proof that ERL implies the Paris-Harrington principle PH. The present authors give a model-theoretic proof of unprovability of ERL in PA, and in the process construct a new indicator for cuts satisfying PA. A similar analysis is applied to the KM$$_j$$ principle. The main results about it are summarized in the following corollary, in which the principles PH$$^n$$ and KM$$^n_j$$ are the relativized versions of PH and KM$$_j$$ for the fixed exponent $$n$$. The following statements are equivalent in $$\mathrm{I}\Sigma_1$$: PH$$^{n+1}$$; KM$$^{n+1}$$; KM$$^{n+j}_j$$; 1-Con$$(\mathrm I\Sigma_n)$$.

### MSC:

 03F30 First-order arithmetic and fragments 03B30 Foundations of classical theories (including reverse mathematics) 03C62 Models of arithmetic and set theory
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### References:

 [1] Bovykin, A., “Several proofs of PA-unprovability,” pp. 29-43 in Logic and Its Applications , vol. 380 of Contemporary Mathematics , American Mathematical Society, Providence, 2005. · Zbl 1084.03045 [2] Bovykin, A., “Brief introduction to unprovability,” pp. 38-64 in Logic Colloquium 2006 , Lecture Notes in Logic, Association for Symbolic Logic, Chicago, 2009. [3] Bovykin, A., and A. Weiermann, “The strength of infinitary Ramseyan statements can be accessed by their densities,” to appear in Annals of Pure and Applied Logic . · Zbl 1422.03127 [4] Carlucci, L., G. Lee, and A. Weiermann, “Sharp thresholds for hypergraph regressive Ramsey numbers,” Journal of Combinatorial Theory, Series A , vol. 118 (2011), pp. 558-85. · Zbl 1251.05103 [5] Carlucci, L., G. Lee, and A. Weiermann, “Classifying the phase transition threshold for regressive Ramsey functions,” preprint, 2006. [6] Carlucci, L., and A. Weiermann, “Classifying the phase transition for canonical Ramsey functions,” preprint, 2010. [7] Erdös, P., and R. Rado, “A combinatorial theorem,” Journal of the London Mathematical Society , vol. 25 (1950), pp. 249-55. · Zbl 0038.15301 [8] Kanamori, A., and K. McAloon, “On Gödel incompleteness and finite combinatorics,” Annals of Pure and Applied Logic , vol. 33 (1987), pp. 23-41. · Zbl 0627.03041 [9] Kaye, R., Models of Peano Arithmetic , vol. 15 of Oxford Logic Guides , Oxford University Press, New York, 1991. · Zbl 0744.03037 [10] Lee, G., “Phase transitions in axiomatic thought,” Ph.D. dissertation, University of Münster, Münster, Germany, 2005. · Zbl 1082.03048 [11] Lefmann, H., and V. Rödl, “On canonical Ramsey numbers for complete graphs versus paths,” Journal of Combinatorial Theory, Series B , vol. 58 (1993), pp. 1-13. · Zbl 0794.05088 [12] Mileti, J. R., “The canonical Ramsey theorem and computability theory,” Transactions of the American Mathematical Society , vol. 360 (2008), pp. 1309-40. · Zbl 1135.03014 [13] Mills, G., “A tree analysis of unprovable combinatorial statements,” pp. 248-311 in Model Theory of Algebra and Arithmetic (Karpacz, Poland, 1978) , vol. 834 of Lecture Notes in Mathematics , Springer, Berlin, 1980. · Zbl 0472.05019 [14] Paris, J., and L. Harrington, “A mathematical incompleteness in Peano arithmetic,” pp. 1133-42 in Handbook of Mathematical Logic , edited by J. Barwise, vol. 90 of Studies in Logic and the Foundations of Mathematics , North-Holland, Amsterdam, 1977. [15] Ramsey, F. P., “On a problem of formal logic,” Proceedings of the London Mathematical Society , vol. 30 (1930), pp. 264-86. · JFM 55.0032.04 [16] Weiermann, A., “An application of graphical enumeration to PA,” Journal of Symbolic Logic , vol. 68 (2003), pp. 5-16. · Zbl 1041.03045 [17] Weiermann, A., “A classification of rapidly growing Ramsey functions,” Proceedings of the American Mathematical Society , vol. 132 (2004), pp. 553-61. · Zbl 1041.03044 [18] Weiermann, A., “Analytic combinatorics, proof-theoretic ordinals, and phase transitions for independence results, Annals of Pure and Applied Logic , vol. 136 (2005), pp. 189-218. · Zbl 1090.03028 [19] Weiermann, A., and W. Van Hoof, “Sharp phase transition thresholds for the Paris Harrington Ramsey numbers for a fixed dimension,” Proceedings of the American Mathematical Society , vol. 140 (2012), pp. 2913-27. · Zbl 1291.03113
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