Ramsey methods in analysis

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.

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.

Discrete Mathematics, 1982

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.

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.

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.

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 ).

Advances in Mathematics, 1992

We state and prove a theorem in exterior algebra which is an analogue of Ramsey's theorem in combinatorics, with vector spaces and alternating multilinear maps taking the roles played by sets and colorings. Our result can also be formulated as a theorem about the geometry of Grassmannians. The infinite-dimensional version of the theorem admits a quantum mechanical interpretation, and implies an interesting fact about operators on Hilbert space.

2010

Given a fixed integer $n$, we prove Ramsey-type theorems for the classes of all finite ordered $n$-colorable graphs, finite $n$-colorable graphs, finite ordered $n$-chromatic graphs, and finite $n$-chromatic graphs.

We consider the numbers associated with Ramsey's theorem as it pertains to partitions of the pairs of elements of a set into two classes. Our purpose is to give a unified development of enumerative techniques which give sharp upper bounds on these numbers and to give constructive methods for partitions to determine lower bounds on these numbers. Explicit computations include the values of R(3, 6) and R(3, 7) among others. Our computational techniques yield the upper bound R(x, y) < cyX-llog log y/log y for x ~> 3.