2016 Phase Transition Results for Three Ramsey-Like Theorems
Florian Pelupessy
Notre Dame J. Formal Logic 57(2): 195-207 (2016). DOI: 10.1215/00294527-3452807
Abstract

We classify a sharp phase transition threshold for Friedman’s finite adjacent Ramsey theorem. We extend the method for showing this result to two previous classifications involving Ramsey theorem variants: the Paris–Harrington theorem and the Kanamori–McAloon theorem. We also provide tools to remove ad hoc arguments from the proofs of phase transition results as much as currently possible.

## References

1.

[1] Arai, T., “Introduction to proof theory,” lecture notes,  http://kurt.scitec.kobe-u.ac.jp/~arai/.[1] Arai, T., “Introduction to proof theory,” lecture notes,  http://kurt.scitec.kobe-u.ac.jp/~arai/.

2.

[2] Buchholz, W., “Beweistheorie,” lecture notes,  http://www.mathematik.uni-muenchen.de/~buchholz/.[2] Buchholz, W., “Beweistheorie,” lecture notes,  http://www.mathematik.uni-muenchen.de/~buchholz/.

3.

[3] Buss, S. R., ed., Handbook of Proof Theory, vol. 137 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1998. MR1640324[3] Buss, S. R., ed., Handbook of Proof Theory, vol. 137 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1998. MR1640324

4.

[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. MR2739504 10.1016/j.jcta.2010.08.004[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. MR2739504 10.1016/j.jcta.2010.08.004

5.

[5] Erdős, P., and R. Rado, “Combinatorial theorems on classifications of subsets of a given set,” Proceedings of the London Mathematical Society (3), vol. 2 (1952), pp. 417–39. MR65615[5] Erdős, P., and R. Rado, “Combinatorial theorems on classifications of subsets of a given set,” Proceedings of the London Mathematical Society (3), vol. 2 (1952), pp. 417–39. MR65615

6.

7.

[7] Friedman, H. M., and F. Pelupessy, “Independence of Ramsey theorem variants using $\varepsilon_{0}$,” preprint,  http://cage.ugent.be/~pelupessy/ARPH.pdfMR3430859 10.1090/proc12759[7] Friedman, H. M., and F. Pelupessy, “Independence of Ramsey theorem variants using $\varepsilon_{0}$,” preprint,  http://cage.ugent.be/~pelupessy/ARPH.pdfMR3430859 10.1090/proc12759

8.

[8] Graham, R. L., B. L. Rothschild, and J. H. Spencer, Ramsey Theory, 2nd ed., Wiley, New York, 1990. MR1044995[8] Graham, R. L., B. L. Rothschild, and J. H. Spencer, Ramsey Theory, 2nd ed., Wiley, New York, 1990. MR1044995

9.

[9] Kanamori, A., and K. McAloon, “On Gödel incompleteness and finite combinatorics,” Annals of Pure and Applied Logic, vol. 33 (1987), pp. 23–41. MR870685 10.1016/0168-0072(87)90074-1[9] Kanamori, A., and K. McAloon, “On Gödel incompleteness and finite combinatorics,” Annals of Pure and Applied Logic, vol. 33 (1987), pp. 23–41. MR870685 10.1016/0168-0072(87)90074-1

10.

[10] Ketonen, J., and R. Solovay, “Rapidly growing Ramsey functions,” Annals of Mathematics (2), vol. 113 (1981), pp. 267–314. MR607894 10.2307/2006985[10] Ketonen, J., and R. Solovay, “Rapidly growing Ramsey functions,” Annals of Mathematics (2), vol. 113 (1981), pp. 267–314. MR607894 10.2307/2006985

11.

[11] Lee, G., “Phase transitions in axiomatic thought,” Ph.D. dissertation, University of Münster, Münster, Germany, 2005.[11] Lee, G., “Phase transitions in axiomatic thought,” Ph.D. dissertation, University of Münster, Münster, Germany, 2005.

12.

[12] Loebl, M., and J. Nešetřil, “An unprovable Ramsey-type theorem,” Proceedings of the American Mathematical Society, vol. 116 (1992), pp. 819–24. MR1095225 10.1090/S0002-9939-1992-1095225-4[12] Loebl, M., and J. Nešetřil, “An unprovable Ramsey-type theorem,” Proceedings of the American Mathematical Society, vol. 116 (1992), pp. 819–24. MR1095225 10.1090/S0002-9939-1992-1095225-4

13.

[13] Paris, J., and L. Harrington, “A mathematical incompleteness in Peano arithmetic,” pp. 1133–42 in Handbook for Mathematical Logic, edited by J. Barwise, vol. 90 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1977. MR457132[13] Paris, J., and L. Harrington, “A mathematical incompleteness in Peano arithmetic,” pp. 1133–42 in Handbook for Mathematical Logic, edited by J. Barwise, vol. 90 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1977. MR457132

14.

[14] Pohlers, W., Proof Theory: An Introduction, vol. 1407 of Lecture Notes in Mathematics, Springer, Berlin, 1989. MR1026933[14] Pohlers, W., Proof Theory: An Introduction, vol. 1407 of Lecture Notes in Mathematics, Springer, Berlin, 1989. MR1026933

15.

[15] Weiermann, A., “A classification of rapidly growing Ramsey functions,” Proceedings of the American Mathematical Society, vol. 132 (2004), pp. 553–61. MR2022381 10.1090/S0002-9939-03-07086-2[15] Weiermann, A., “A classification of rapidly growing Ramsey functions,” Proceedings of the American Mathematical Society, vol. 132 (2004), pp. 553–61. MR2022381 10.1090/S0002-9939-03-07086-2

16.

[16] Weiermann, A., “Webpage on phase transitions,” preprint,  http://cage.ugent.be/~weierman//phase.html.[16] Weiermann, A., “Webpage on phase transitions,” preprint,  http://cage.ugent.be/~weierman//phase.html.
Copyright © 2016 University of Notre Dame
Florian Pelupessy "Phase Transition Results for Three Ramsey-Like Theorems," Notre Dame Journal of Formal Logic 57(2), 195-207, (2016). https://doi.org/10.1215/00294527-3452807
Received: 20 March 2013; Accepted: 29 October 2013; Published: 2016
JOURNAL ARTICLE
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Vol.57 • No. 2 • 2016