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2006, Archive for Mathematical Logic
We consider the relation between versions of Ramsey's Theorem and König's Infinity Lemma, in the absence of the axiom of choice.
In combinatorics, Ramsey Theory considers partitions of some mathematical objects and asks the following question: how large must the original object be in order to guarantee that at least one of the parts in the partition exhibits some property? Perhaps the most familiar case is the well-known Pigeonhole Principle: if m pigeonholes house p pigeons where p m, then one of the pigeonholes must contain multiple pigeons. Conversely, the number of pigeons must exceed m in order to guarantee this property.
Proceedings of the American Mathematical Society, 1992
We present a new proof of the Paris-Harrington unprovable (in PA) version of Ramsey's theorem. This also yields a particularly short proof of the Ketonen-Solovay result on rapidly growing Ramsey functions.
In [21], Frank Plumpton Ramsey proved what has turned out to be a remarkable and important theorem which is now known as Ramsey's theorem. This result is a generalization of the pigeonhole principle and can now be seen as part of a family of theorems of the same flavour. These Ramsey-type theorems all have the common feature that they assert, in some precise combinatorial way, that if we deal with large enough sets of numhers, there will be some well behaved fragment in the set. In Harrington's words, Ramsey-type theorems assert that complete chaos is impossible. Ramsey-type theorems have turned out to be very important in a number of branches of mathematics. In this paper we shall survey a number of basic Ramsey-type theorems, and we will then look at a selection of applications of Ramsey-type theorems and Ramsey-type ideas. In the applications we will concentrate on graph theory, logic and complexity theory. Proofs will mostly not be given in detail, but it is hoped that the reader will gain some appreciation of the usefulness and importance of the beautiful area of asymptotic combinatorics.
2005
Abstract We conduct a model-theoretic investigation of three infinitary ramseyan statements: Ramsey Theorem for pairs and two colours (RT2 2), Canonical Ramsey Theorem for pairs (CRT2) and Regressive Ramsey Theorem for pairs (RegRT2). We prove theorems that approximate the logical strength of these principles by the strength of their finite iterations known as density principles. We then investigate their logical strength using strong initial segments of models of Peano Arithmetic, in the spirit of Paris-Kirby results.
Duke Mathematical Journal, 2013
Ramsey's theorem, in the version of Erdős and Szekeres, states that every 2-coloring of the edges of the complete graph on {1, 2,. .. , n} contains a monochromatic clique of order 1 2 log n. In this paper, we consider two well-studied extensions of Ramsey's theorem. Improving a result of Rödl, we show that there is a constant c > 0 such that every 2-coloring of the edges of the complete graph on {2, 3, ..., n} contains a monochromatic clique S for which the sum of 1/ log i over all vertices i ∈ S is at least c log log log n. This is tight up to the constant factor c and answers a question of Erdős from 1981. Motivated by a problem in model theory, Väänänen asked whether for every k there is an n such that the following holds. For every permutation π of 1,. .. , k − 1, every 2-coloring of the edges of the complete graph on {1, 2,. .. , n} contains a monochromatic clique a 1 <. .. < a k with a π(1)+1 − a π(1) > a π(2)+1 − a π(2) >. .. > a π(k−1)+1 − a π(k−1). That is, not only do we want a monochromatic clique, but the differences between consecutive vertices must satisfy a prescribed order. Alon and, independently, Erdős, Hajnal and Pach answered this question affirmatively. Alon further conjectured that the true growth rate should be exponential in k. We make progress towards this conjecture, obtaining an upper bound on n which is exponential in a power of k. This improves a result of Shelah, who showed that n is at most double-exponential in k.
Transactions of the American Mathematical Society, 2008
We show that Ramsey theory, a domain presently conceived to guarantee the existence of large homogeneous sets for partitions on k-tuples of words (for every natural number k) over a finite alphabet, can be extended to one for partitions on Schreier-type sets of words (of every countable ordinal). Indeed, we establish an extension of the partition theorem of Carlson about words and of the (more general) partition theorem of Furstenberg-Katznelson about combinatorial subspaces of the set of words (generating from k-tuples of words for any fixed natural number k) into a partition theorem about combinatorial subspaces (generating from Schreier-type sets of words of order any fixed countable ordinal). Furthermore, as a result we obtain a strengthening of Carlson's infinitary Nash-Williams type (and Ellentuck type) partition theorem about infinite sequences of variable words into a theorem, in which an infinite sequence of variable words and a binary partition of all the finite sequences of words, one of whose components is, in addition, a tree, are assumed, concluding that all the Schreier-type finite reductions of an infinite reduction of the given sequence have a behavior determined by the Cantor-Bendixson ordinal index of the tree-component of the partition, falling in the tree-component above that index and in its complement below it.
2019
The Borel covering property, introduced a century ago by E. Borel, is intimately connected with Ramsey theory, initiated ninety years ago in an influential paper of F.P. Ramsey. The current state of knowledge about the connection between the Borel covering property and Ramsey theory is outlined in this paper. Initially the connection is established for the situation when the set with the Borel covering property is a proper subset of a $\sigma$-compact uniform space. Then the connection is explored for a stronger covering property introduced by Rothberger. After establishing the fact that in this case several landmark Ramseyan theorems are characteristic of this stronger covering property, the case when the space with this stronger covering property is in fact $\sigma$-compact is explored.
König [1926] includes a result subsequently called König 's Infinity Lemma. König [1927] includes a graph theoretic formulation: an infinite, locally finite and connected (undirected) graph includes an infinite path. Contemporary applications of the infinity lemma in logic frequently refer to a consequence of the infinity lemma: an infinite, locally finite (undirected) tree with a root has a infinite branch. This tree lemma can be traced to Beth [1955]. It is argued that Beth independently discovered the tree lemma in the early 50's and that it was later recognized among logicians that Beth's result was a consequence of the infinity lemma. It appears that the question of whether or not the two lemmas are equivalent was not raised in the logic literature. The equivalence of these lemmas is an easy consequence of a well known result in graph theory: every connected, locally finite graph has among its partial subgraphs a spanning tree.
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Pacific Journal of Mathematics, 1980
The Mathematical Intelligencer, 2007
BASEL, BIRKHAUSER, 2005, PP 257, C38, ISBN 3-7643-7264-8 REVIEWED BY HANS-PETER A. KLINZI Ihe book introduces the reader i ] to sophisticated Ramsey-theoretic I methods that have recently been used in the theo W of Banach spaces.
The Journal of Symbolic Logic, 2007
We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is wellknown that Ramsey's Theorem for pairs () splits into a stable version () and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for and ). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL0, DNR, and BΣ2. We show, for instance, that WKL0 is ...
Journal of Symbolic Logic, 2014
We characterize the computational content and the proof-theoretic strength of a Ramseytype theorem for bi-colorings of so-called exactly large sets. An exactly large set is a set X ⊂ N such that card(X) = min(X) + 1. The theorem we analyze is as follows. For every infinite subset M of N, for every coloring C of the exactly large subsets of M in two colors, there exists and infinite subset L of M such that C is constant on all exactly large subsets of L. This theorem is essentially due to Pudlàk and Rödl and independently to Farmaki. We prove that -over Computable Mathematics -this theorem is equivalent to closure under the ω Turing jump (i.e., under arithmetical truth). Natural combinatorial theorems at this level of complexity are rare. Our results give a complete characterization of the theorem from the point of view of Computable Mathematics and of the Proof Theory of Arithmetic. This nicely extends the current knowledge about the strength of Ramsey Theorem. We also show that analogous results hold for a related principle based on the Regressive Ramsey Theorem. In addition we give a further characterization in terms of truth predicates over Peano Arithmetic. We conjecture that analogous results hold for larger ordinals.
Arxiv preprint arXiv:1011.4263, 2010
We give a parametrization with perfect subsets of 2 ∞ of the abstract Ramsey theorem (see ). Our main tool is an adaptation, to a more general context of Ramsey spaces, of the techniques developed in [8] by J. G. Mijares in order to obtain the corresponding result within the context of topological Ramsey spaces. This tool is inspired by Todorcevic's abstract version of the combinatorial forcing introduced by Galvin and Prikry in , and also by the parametrized version of this combinatorial technique, developed in by Pawlikowski. The main result obtained in this paper (theorem 5 below) turns out to be a generalization of the parametrized Ellentuck theorem of , and it yields as corollary that the family of perfectly Ramsey sets corresponding to a given Ramsey space is closed under the Souslin operation. This enabled us to prove a parametrized version of the infinite dimensional Hales-Jewett theorem (see ).
The study of partition properties of the set of real num- bers in several of its dierent presentations has been a very active field of research with interesting and sometimes surprising results. These partition properties are of the following form: if the set of real num- bers (or a related space) is partitioned into a finite collection of pieces, there is a "large" collection of reals contained in one of the pieces. Dif- ferent notions of largeness have been considered, and they give rise to properties of varied combinatorial character. The interplay be- tween metamathematical questions and combinatorial problems has been present throughout the development of the theory. Most of these properties are, in their full generality, inconsistent with the axiom of choice, but versions of them where only partitions into simple pieces are considered, for example, Borel pieces, can be proved to be true. Nevertheless, the unrestricted versions are consistent with weak forms of the ax...
Journal of Mathematical Logic, 2013
In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from RT, Ramsey's Theorem for Pairs. These principles tend to fall outside of the "big five" systems of reverse mathematics and a complicated picture of subsystems below RT has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that ADS is not equivalent to CAC and that EM is not equivalent to RT.
Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1992
We give a simple game-theoretic proof of Silver's theorem that every analytic set is Ramsey. A set P of subsets of o is called Ramsey if there exists an infinite set H such that either all infinite subsets of H are in P or all out of P. Our proof clarifies a strong connection between the Ramsey property of partitions and the determinacy of infinite games.
Bulletin of the Polish Academy of Sciences Mathematics
In ZFA (Zermelo-Fraenkel set theory with the Axiom of Extensionality weakened to allow the existence of atoms), we prove that the strength of the proposition EDM ("If G = (VG, EG) is a graph such that VG is uncountable, then for every coloring f : [VG] 2 → {0, 1} either there is an uncountable set monochromatic in color 0, or there is a countably infinite set monochromatic in color 1") is strictly between DC ℵ 1 (where DC ℵ 1 is Dependent Choices for ℵ1, a weak choice form stronger than Dependent Choices (DC)) and Kurepa's principle ("Any partially ordered set such that all of its antichains are finite and all of its chains are countable is countable"). Among other new results, we study the relations of EDM to BPI (Boolean Prime Ideal Theorem), RT (Ramsey's theorem), De Bruijn-Erdős' theorem for n-colorings, König's lemma and several other weak choice forms. Moreover, we answer a part of a question raised by Lajos Soukup.
arXiv: Logic, 2017
Ramsey theory and forcing have a symbiotic relationship. At the RIMS Symposium on Infinite Combinatorics and Forcing Theory in 2016, the author gave three tutorials on Ramsey theory in forcing. The first two tutorials concentrated on forcings which contain dense subsets forming topological Ramsey spaces. These forcings motivated the development of new Ramsey theory, which then was applied to the generic ultrafilters to obtain the precise structure Rudin-Keisler and Tukey orders below such ultrafilters. The content of the first two tutorials has appeared in an expository article submitted to the SEALS 2016 Proceedings. The third tutorial concentrated on uses of forcing to prove Ramsey theorems for trees which are applied to determine big Ramsey degrees of homogeneous relational structures. This is the focus of this paper.
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