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Sharp phase transition thresholds for the Paris Harrington Ramsey numbers for a fixed dimension

Andreas Weiermann (UGent) and Wim Van Hoof (UGent)
Author
Organization
Project
phase transitions in logic and combinatorics
Abstract
This article is concerned with investigations on a phase transition which is related to the (finite) Ramsey theorem and the Paris-Harrington theorem. For a given number-theoretic function g, let R-c(d)(g)(k) be the least natural number R such that for all colourings P of the d-element subsets of {0, . . . ,R - 1} with at most c colours there exists a subset H of {0, . . . , R - 1} such that P has constant value on all d-element subsets of H and such that the cardinality of H is not smaller than max{k, g(min(H))}. If g is a constant function with value e, then R-c(d)(g)(k) is equal to the usual Ramsey number R-c(d)(max{e, k}); and if g is the identity function, then R-c(d)(g)(k) is the corresponding Paris-Harrington number, which typically is much larger than R-c(d)(k). In this article we give for all d >= 2 a sharp classification of the functions g for which the function m bar right arrow R-m(d)(g)(m) grows so quickly that it is no longer provably total in the subsystem of Peano arithmetic, where the induction scheme is restricted to formulas with at most (d - 1)-quantifiers. Such a quick growth will in particular happen for any function g growing at least as fast as i bar right arrow epsilon . log(. . . (log(i). . .)(sic)(d-1)-times (where epsilon > 0 is fixed) but not for the function g(i) = 1/log*(i) . log(. . . (log( i). . .).(sic)(d-1)-times (Here log* denotes the functional inverse of the tower function.) To obtain such results arid even sharper bounds we employ certain suitable transfinite iterations of nonconstructive lower bound functions for Ramsey numbers. Thereby we improve certain results from the article A classification of rapidly growing Ramsey numbers (PAMS 132 (2004), 553-561) of the first author, which were obtained by employing constructive ordinal partitions.
Keywords
Ramsey theorem, rapidly growing Ramsey functions, Peano arithmetic, fast growing hierarchies

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Citation

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Chicago
Weiermann, Andreas, and Wim Van Hoof. 2012. “Sharp Phase Transition Thresholds for the Paris Harrington Ramsey Numbers for a Fixed Dimension.” Proceedings of the American Mathematical Society 140 (8): 2913–2927.
APA
Weiermann, A., & Van Hoof, W. (2012). Sharp phase transition thresholds for the Paris Harrington Ramsey numbers for a fixed dimension. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 140(8), 2913–2927.
Vancouver
1.
Weiermann A, Van Hoof W. Sharp phase transition thresholds for the Paris Harrington Ramsey numbers for a fixed dimension. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY. 2012;140(8):2913–27.
MLA
Weiermann, Andreas, and Wim Van Hoof. “Sharp Phase Transition Thresholds for the Paris Harrington Ramsey Numbers for a Fixed Dimension.” PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 140.8 (2012): 2913–2927. Print.
@article{1972866,
  abstract     = {This article is concerned with investigations on a phase transition which is related to the (finite) Ramsey theorem and the Paris-Harrington theorem. For a given number-theoretic function g, let R-c(d)(g)(k) be the least natural number R such that for all colourings P of the d-element subsets of \{0, . . . ,R - 1\} with at most c colours there exists a subset H of \{0, . . . , R - 1\} such that P has constant value on all d-element subsets of H and such that the cardinality of H is not smaller than max\{k, g(min(H))\}. If g is a constant function with value e, then R-c(d)(g)(k) is equal to the usual Ramsey number R-c(d)(max\{e, k\}); and if g is the identity function, then R-c(d)(g)(k) is the corresponding Paris-Harrington number, which typically is much larger than R-c(d)(k). In this article we give for all d {\textrangle}= 2 a sharp classification of the functions g for which the function m bar right arrow R-m(d)(g)(m) grows so quickly that it is no longer provably total in the subsystem of Peano arithmetic, where the induction scheme is restricted to formulas with at most (d - 1)-quantifiers. Such a quick growth will in particular happen for any function g growing at least as fast as i bar right arrow epsilon . log(. . . (log(i). . .)(sic)(d-1)-times (where epsilon {\textrangle} 0 is fixed) but not for the function g(i) = 1/log*(i) . log(. . . (log( i). . .).(sic)(d-1)-times (Here log* denotes the functional inverse of the tower function.) To obtain such results arid even sharper bounds we employ certain suitable transfinite iterations of nonconstructive lower bound functions for Ramsey numbers. Thereby we improve certain results from the article A classification of rapidly growing Ramsey numbers (PAMS 132 (2004), 553-561) of the first author, which were obtained by employing constructive ordinal partitions.},
  author       = {Weiermann, Andreas and Van Hoof, Wim},
  issn         = {0002-9939},
  journal      = {PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY},
  keyword      = {Ramsey theorem,rapidly growing Ramsey functions,Peano arithmetic,fast growing hierarchies},
  language     = {eng},
  number       = {8},
  pages        = {2913--2927},
  title        = {Sharp phase transition thresholds for the Paris Harrington Ramsey numbers for a fixed dimension},
  url          = {http://dx.doi.org/10.1090/S0002-9939-2011-11121-3},
  volume       = {140},
  year         = {2012},
}

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