737 research outputs found
The weakness of being cohesive, thin or free in reverse mathematics
Informally, a mathematical statement is robust if its strength is left
unchanged under variations of the statement. In this paper, we investigate the
lack of robustness of Ramsey's theorem and its consequence under the frameworks
of reverse mathematics and computable reducibility. To this end, we study the
degrees of unsolvability of cohesive sets for different uniformly computable
sequence of sets and identify different layers of unsolvability. This analysis
enables us to answer some questions of Wang about how typical sets help
computing cohesive sets.
We also study the impact of the number of colors in the computable
reducibility between coloring statements. In particular, we strengthen the
proof by Dzhafarov that cohesiveness does not strongly reduce to stable
Ramsey's theorem for pairs, revealing the combinatorial nature of this
non-reducibility and prove that whenever is greater than , stable
Ramsey's theorem for -tuples and colors is not computably reducible to
Ramsey's theorem for -tuples and colors. In this sense, Ramsey's
theorem is not robust with respect to his number of colors over computable
reducibility. Finally, we separate the thin set and free set theorem from
Ramsey's theorem for pairs and identify an infinite decreasing hierarchy of
thin set theorems in reverse mathematics. This shows that in reverse
mathematics, the strength of Ramsey's theorem is very sensitive to the number
of colors in the output set. In particular, it enables us to answer several
related questions asked by Cholak, Giusto, Hirst and Jockusch.Comment: 31 page
Open questions about Ramsey-type statements in reverse mathematics
Ramsey's theorem states that for any coloring of the n-element subsets of N
with finitely many colors, there is an infinite set H such that all n-element
subsets of H have the same color. The strength of consequences of Ramsey's
theorem has been extensively studied in reverse mathematics and under various
reducibilities, namely, computable reducibility and uniform reducibility. Our
understanding of the combinatorics of Ramsey's theorem and its consequences has
been greatly improved over the past decades. In this paper, we state some
questions which naturally arose during this study. The inability to answer
those questions reveals some gaps in our understanding of the combinatorics of
Ramsey's theorem.Comment: 15 page
Reverse mathematics and infinite traceable graphs
This paper falls within the general program of investigating the proof
theoretic strength (in terms of reverse mathematics) of combinatorial
principals which follow from versions of Ramsey's theorem. We examine two
statements in graph theory and one statement in lattice theory proved by
Galvin, Rival and Sands \cite{GRS:82} using Ramsey's theorem for 4-tuples. Our
main results are that the statements concerning graph theory are equivalent to
Ramsey's theorem for 4-tuples over \RCA while the statement concerning
lattices is provable in \RCA.
Revised 12/2010. To appear in Archive for Mathematical Logi
Controlling iterated jumps of solutions to combinatorial problems
Among the Ramsey-type hierarchies, namely, Ramsey's theorem, the free set,
the thin set and the rainbow Ramsey theorem, only Ramsey's theorem is known to
collapse in reverse mathematics. A promising approach to show the strictness of
the hierarchies would be to prove that every computable instance at level n has
a low_n solution. In particular, this requires effective control of iterations
of the Turing jump. In this paper, we design some variants of Mathias forcing
to construct solutions to cohesiveness, the Erdos-Moser theorem and stable
Ramsey's theorem for pairs, while controlling their iterated jumps. For this,
we define forcing relations which, unlike Mathias forcing, have the same
definitional complexity as the formulas they force. This analysis enables us to
answer two questions of Wei Wang, namely, whether cohesiveness and the
Erdos-Moser theorem admit preservation of the arithmetic hierarchy, and can be
seen as a step towards the resolution of the strictness of the Ramsey-type
hierarchies.Comment: 32 page
Ramsey's Theorem for Pairs and Colors as a Sub-Classical Principle of Arithmetic
The purpose is to study the strength of Ramsey's Theorem for pairs restricted
to recursive assignments of -many colors, with respect to Intuitionistic
Heyting Arithmetic. We prove that for every natural number , Ramsey's
Theorem for pairs and recursive assignments of colors is equivalent to the
Limited Lesser Principle of Omniscience for formulas over Heyting
Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is
equivalent to: for every recursively enumerable infinite -ary tree there is
some and some branch with infinitely many children of index .Comment: 17 page
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