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On uniform relationships between combinatorial problems
The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic. In this setting, called reverse mathematics, one investigates whichExpand
The polarized Ramsey’s theorem
TLDR
This investigation yields a new characterization of Ramsey's theorem in all exponents, and produces several combinatorial principles which, modulo bounding for $$Sigma^0_2}$$ formulas, lie (possibly not strictly) between Ramsey’s theorem for pairs and the stable Ramsey�'s theorem for pair. Expand
Combinatorics and graph theory
Graph Theory: Introductory Concepts.- Trees.- Planarity.- Colorings.- Matchings.- Ramsey Theory.- References Combinatorics: Three Basic Problems.- Binomial Coefficients.- The Principle of InclusionExpand
Weak Comparability of Well Orderings and Reverse Mathematics
TLDR
The main goal of this paper is to give a detailed exposition of this result that any two countable well orderings are weakly comparable if there is an order preserving injection of one into the other. Expand
Reverse Mathematics and Ordinal Exponentiation
  • J. Hirst
  • Mathematics, Computer Science
  • Ann. Pure Appl. Log.
  • 10 February 1994
TLDR
This paper provides empirical support for Simpson's claim that "ATRo is the weakest set of axioms which permits the development of a decent theory of countable ordinals" and analyzes the provability of statements about countable well­ orderings within weak subsystems of second-order arithmetic. Expand
Combinatorics and Graph Theory, Second Edition
Generics for computable Mathias forcing
TLDR
It is shown that if G is any n-generic with n ≥ 2 then it satisfies the jump property G ( n − 1 ) ≡ T G ′ ⊕ ∅ ( n ) , and so cannot have even Cohen 1-generic degree. Expand
Reverse Mathematics and Uniformity in Proofs without Excluded Middle
TLDR
If a $\Pi^1_2$ sentence of a certain form is provable using E-HA${}^\omega$ along with the axiom of choice and an independence of premise principle, the sequential form of the statement is Provable in the classical system RCA. Expand
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