Coloring trees in reverse mathematics

The tree theorem for pairs ($\mathsf{TT}^2_2$), first introduced by Chubb, Hirst, and McNicholl, asserts that given a finite coloring of pairs of comparable nodes in the full binary tree $2^{<\omega}$, there is a set of nodes isomorphic to $2^{<\omega}$ which is homogeneous for the coloring. This is a generalization of the more familiar Ramsey's theorem for pairs ($\mathsf{RT}^2_2$), which has been studied extensively in computability theory and reverse mathematics. We answer a longstanding open question about the strength of $\mathsf{TT}^2_2$, by showing that this principle does not imply the arithmetic comprehension axiom ($\mathsf{ACA}_0$) over the base system, recursive comprehension axiom ($\mathsf{RCA}_0$), of second-order arithmetic. In addition, we give a new and self-contained proof of a recent result of Patey that $\mathsf{TT}^2_2$ is strictly stronger than $\mathsf{RT}^2_2$. Combined, these results establish $\mathsf{TT}^2_2$ as the first known example of a natural combinatorial principle to occupy the interval strictly between $\mathsf{ACA}_0$ and $\mathsf{RT}^2_2$. The proof of this fact uses an extension of the bushy tree forcing method, and develops new techniques for dealing with combinatorial statements formulated on trees, rather than on $\omega$.Comment: 25 page

The logical strength of Büchi's decidability theorem

We study the strength of axioms needed to prove various results related to automata on infinite words and Büchi's theorem on the decidability of the MSO theory of (N, less_or_equal). We prove that the following are equivalent over the weak second-order arithmetic theory RCA: 1. Büchi's complementation theorem for nondeterministic automata on infinite words, 2. the decidability of the depth-n fragment of the MSO theory of (N, less_or_equal), for each n greater than 5, 3. the induction scheme for Sigma^0_2 formulae of arithmetic. Moreover, each of (1)-(3) is equivalent to the additive version of Ramsey's Theorem for pairs, often used in proofs of (1); each of (1)-(3) implies McNaughton's determinisation theorem for automata on infinite words; and each of (1)-(3) implies the "bounded-width" version of König's Lemma, often used in proofs of McNaughton's theorem

The strength of the tree theorem for pairs in reverse mathematics

International audienceNo natural principle is currently known to be strictly between the arithmetic comprehension axiom (ACA0) and Ramsey's theorem for pairs (RT 2 2) in reverse mathematics. The tree theorem for pairs (TT 2 2) is however a good candidate. The tree theorem states that for every finite coloring over tuples of comparable nodes in the full binary tree, there is a monochromatic subtree isomorphic to the full tree. The principle TT 2 2 is known to lie between ACA0 and RT 2 2 over RCA0, but its exact strength remains open. In this paper, we prove that RT 2 2 together with weak König's lemma (WKL0) does not imply TT 2 2 , thereby answering a question of Montálban. This separation is a case in point of the method of Lerman, Solomon and Towsner for designing a computability-theoretic property which discriminates between two statements in reverse mathematics. We therefore put the emphasis on the different steps leading to this separation in order to serve as a tutorial for separating principles in reverse mathematics

Coloring trees in reverse mathematics

International audienceThe tree theorem for pairs (TT 2 2), first introduced by Chubb, Hirst, and McNicholl, asserts that given a finite coloring of pairs of comparable nodes in the full binary tree 2 <ω , there is a set of nodes isomorphic to 2 <ω which is homogeneous for the coloring. This is a generalization of the more familiar Ramsey's theorem for pairs (RT 2 2), which has been studied extensively in computability theory and reverse mathematics. We answer a longstanding open question about the strength of TT 2 2 , by showing that this principle does not imply the arithmetic comprehension axiom (ACA 0) over the base system, recursive comprehension axiom (RCA 0), of second-order arithmetic. In addition , we give a new and self-contained proof of a recent result of Patey that TT 2 2 is strictly stronger than RT 2 2. Combined, these results establish TT 2 2 as the first known example of a natural combinatorial principle to occupy the interval strictly between ACA 0 and RT 2 2. The proof of this fact uses an extension of the bushy tree forcing method, and develops new techniques for dealing with combinatorial statements formulated on trees, rather than on ω

An algorithmic approach for multi-color Ramsey graphs

The classical Ramsey number R(r1,r2,...,rm) is defined to be the smallest integer n such that no matter how the edges of Kn are colored with the m colors, 1, 2, 3, . . . ,m, there exists some color i such that there is a complete subgraph of size ri, all of whose edges are of color i. The problem of determining Ramsey numbers is known to be very difficult and is usually split into two problems, finding upper and lower bounds. Lower bounds can be obtained by the construction of a, so called, Ramsey graph. There are many different methods to construct Ramsey graphs that establish lower bounds. In this thesis mathematical and computational methods are exploited to construct Ramsey graphs. It was shown that the problem of checking that a graph coloring gives a Ramsey graph is NP-complete. Hence it is almost impossible to find a polynomial time algorithm to construct Ramsey graphs by searching and checking. Consequently, a method such as backtracking with good pruning techniques should be used. Algebraic methods were developed to enable such a backtrack search to be feasible when symmetry is assumed. With the algorithm developed in this thesis, two new lower bounds were established: R(3,3,5)≥45 and R(3,4,4)≥55. Other best known lower bounds were matched, such as R(3,3,4)≥30. The Ramsey graphs giving these lower bounds were analyzed and their full symmetry groups were determined. In particular it was shown that there are unique cyclic graphs up to isomorphism giving R(3,3,4)≥30 and R(3,4,4)≥55, and 13 non-isomorphic cyclic graphs giving R(3,3,5)≥45

Partition Theorems for Spaces of Variable Words

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135422/1/plms0449.pd

Reverse Mathematics of Ramsey\u27s Theorem

Reverse mathematics aims to determine which set theoretic axioms are necessary to prove the theorems outside of the set theory. Since the 1970’s, there has been an interest in applying reverse mathematics to study combinatorial principles like Ramsey’s theorem to analyze its strength and relation to other theorems. Ramsey’s theorem for pairs states that for any infinite complete graph with a finite coloring on edges, there is an infinite subset of nodes all of whose edges share one color. In this thesis, we introduce the fundamental terminology and techniques for reverse mathematics, and demonstrate their use in proving Kőnig\u27s lemma and Ramsey\u27s theorem over RCA0

Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination

This paper is intended to provide an introduction to cut elimination which is accessible to a broad mathematical audience. Gentzen's cut elimination theorem is not as well known as it deserves to be, and it is tied to a lot of interesting mathematical structure. In particular we try to indicate some dynamical and combinatorial aspects of cut elimination, as well as its connections to complexity theory. We discuss two concrete examples where one can see the structure of short proofs with cuts, one concerning feasible numbers and the other concerning "bounded mean oscillation" from real analysis

Truth and Probability

Contains two other essays as well: Further Considerations & Last Papers: Probability and Partial Belief.

A constructive interpretation of Ramsey's theorem via the product of selection functions

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