We show by elementary methods that given any finite partition of the set N of positive integers, there is one cell that is both additively and multiplicatively rich. In particular, this cell must contain a sequence and all of its finite sums, and another sequence and all of its finite products, a fact that was previously known only by utilizing the algebraic structure of the Stone-Cech compactification fiN of N.

We prove a theorem about idempotents in compact semigroups. This theorem gives a new proof of van der Waerden’s theorem on arithmetic progressions as well as the Hales-Jewett theorem. It also gives an infinitary version of the Hales-Jewett theorem which includes results of T. J. Carlson and S. G. Simpson.

We investigate the effect of a variant of Matet forcing on ultrafilters in the ground model and give a characterization of those P –points that survive such forcing, answering a question left open by Blass [4]. We investigate the question of when this variant of Matet forcing can be used to diagonalize small filters without destroying P–points in the ground model. We also deal with the question of generic existence of stable ordered-union ultrafilters.

We discuss methods for identifying random phenomena which are logical consequences of the fact that almost all real numbers are of a high descriptive complexity. These methods enable us to find complexity-theoretic versions of classical results in the theory of random series. In addition, we show how the outcomes which are of high descriptive complexity of random processes bring into realisation the highly reflective (self-similar) nature of various combinatorial configurations. The paper uses methods of recursion theory.

We derive a generalization of a theorem of Raimi proving there is a partition of natural numbers with given densities of classes which meet structured translates of any other class of a partition of natural numbers.

Utilizing ultrafilters on the setN of natural numbers which have certain special properties, we prove some simultaneous generalizations of Ramsey's Theorem and several single dimension Ramsey-type theorems. (For instance, as a very special case, we obtain a simultaneous generalization of Ramsey's Theorem and van der Waerden's Theorem.) Examples are given to show directions in which these results cannot be extended.

Ultrafilters are a tool, originating in mathematical logic and general topology, that has steadily found more and more uses in multiple areas of mathematics, such as combinatorics, dynamics, and algebra, among others. The purpose of this article is to introduce ultrafilters in a friendly manner and present some applications to the branch of combinatorics known as Ramsey theory, culminating with a new ultrafilter-based proof of van der Waerden’s

The main result provide a common generalization for Ramsey-type theorems concerning finite colorings of edge sets of complete graphs with vertices in infinite semigroups. We capture the essence of theorems proved in different fields: for natural numbers due to Milliken–Tylor, Deuber–Hindman, Bergelson–Hindman, for combinatorial covering properties due to Scheepers and Tsaban, and local properties in function spaces due to Scheepers. To this end, we use idempotent ultrafilters in the Čech–Stone compactifications of discrete infinite semigroups and topological games. The research is motivated by the recent breakthrough work of Tsaban about colorings and the Menger covering property. 1. Background 1.1. Colorings and natural numbers. A coloring of a nonempty set X is a function χ : X → {1, . . . , k}, where k is a natural number. Given a coloring χ of a set X , a set A ⊆ X is χ-monochromatic (or just monochromatic, when χ is clear from the context), if there is a color i such that χ(a) = i for all a ∈ A. By the van der Waerden Theorem [37], for each coloring of the set of natural numbers N, there is a monochromatic arithmetic progression of an arbitrarily finite length. In the comprehensive work, Bergelson and Hindman [4] considered families of finite subsets of N with the property that for each coloring of N, there is a monochromatic set in the family. By the result of Hindman [16, Theorem 6.7], a family of finite subsets of N has the above property if and only if there is an ultrafilter on N such that each set in the ultrafilter contains a set in the family. Ultrafilters play an important role in consideration of colorings of N, especially, when an algebraic structure of N is involved. Let S be an infinite semigroup with the discrete topology. Usually, we denote the semigroup operation by +, even if this operation is not commutative. The Čech–Stone compactification, βS, of S is the family of all ultrafilters on S with the topology generated by the sets { p ∈ βS : A ∈ p }, where A ⊆ S. The operation + on S can be extended to the operation + on βS such that for ultrafilters p, q ∈ βS, we have A ∈ p+ q if and only if {

The Erdos sum of reciprocals conjecture is the statement that whenever A is a set of positive integers and ∑∈ A 1/ x = ∞, A contains arbitrarily long arithmetic progressions. It is shown here that this conjecture is equivalent to each of several other statements. Some of these other statements are combinatorial in nature while others are topological-algebraic statements.

Ramsey theory is best defined by example and the classic example of a Ramsey type theorem is the result of van der Waerden: if the integers Z are partitioned into finitely many sets, one of these contains arbitrarily long arithmetic progressions. An equivalent version states: given natural numbers r, 1, 3N(r,l) so that if N > N(r} 1) and the integers {1,2,...,N} are partitioned into r sets, one of these contains an /-term arithmetic progression. Erdos and Turân realized that this result would follow if it were the case that any subset of {1,2,..., N} comprising ON elements contains an /-term arithmetic progression provided N is sufficiently large. This result, conjectured in 1936 [ETI] was proved by E. Szemeredi in 1973 [Szl], and was reproved using ergodic theoretic methods in 1976 [Fui]. The theorems of van den Waerden and Szemeredi illustrate the two sides of Ramsey theory: coloring or partition results, and density results. The latter are clearly stronger than the former, and the proofs are typically more recondite. The role of ergodic theory in density theorems of this type stems from the fact that in a number of situations theorems about patterns found in sets having "density" bounded from below are equivalent to theorems about the "return", or "recurrence", patterns for measure preserving transformations acting on a measure space. It would appear that the ubiquity of certain patterns in the combinatorics of large sets reflects the phenomenon of recurrence in ergodic theoretic contexts and the latter has to be studied to gain insight into the former. We shall be examining three different contexts for recurrence results in ergodic theory. We shall mention the combinatorial (Ramsey theoretic) equivalents of these results, although we shall have to refer the reader elsewhere for the proof of the equivalence. Our purpose here is to display the common features of these recurrence phenomena. We shall find that in each setting there is a notion of rigidity and the notion of a special system constructed from scratch by a finite succession of rigid extension. For these systems (distal and quasi-distal) it will be possible to deduce recurrence properties directly, the key tool being Ramsey theory of the van der Waerden sort. For arbitrary systems we shall find a method to "factor out" the "independent" component, reducing the recurrence property

Previous research extending over a few decades has established that multiplicatively large sets (in any of several interpretations) must have substantial additive structure. We investigate here the question of how much multiplicative structure can be found in additively large sets. For example, we show that any translate of a set of finite sums from an infinite sequence must contain all of the initial products from another infinite sequence. And, as a corollary of a result of Renling Jin, we show that if A and B have positive upper Banach density, then A + B contains all of the initial products from an infinite sequence. We also show that if a set has a complement which is not additively piecewise syndetic, then any translate of that set is both additively and multiplicatively large in several senses.We investigate whether a subset of N with bounded gaps--a syndetic set--must contain arbitrarily long geometric progressions. We believe that we establish that this is a significant open question.

Preliminaries Van der Waerden's theorem Supersets of $AP$ Subsets of $AP$ Other generalizations of $w(k r)$ Arithmetic progressions (mod $m$) Other variations on van der Waerden's theorem Schur's theorem Rado's theorem Other topics Notation Biobliography Index.

In their, by now classical, paper ‘Ramsey’s theorem for n-parameter sets’ (Trans. Amer. Math. Soc. 159 (1971), 257–291) Graham and Rothschild introduced a combinatorial structure which turned out be central in Ramsey theory. In this paper we survey the development related to the structure of Graham-Rothschild parameter sets.

IP* sets and central sets are subsets of ℕ which are known to have rich combinatorial structure. We establish here that structure is significantly richer that was previously known. We also establish that multiplicatively central sets have rich additive structure. The relationship among IP* sets, central sets, and corresponding dynamical notions are also investigated.

Hindman’s theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. Ramsey algebras are structures that satisfy an analogue of Hindman’s theorem. The main purpose of this paper is to introduce Ramsey algebras and to present some basic results of their theory. In particular, we show that every infinite integral domain is not a Ramsey algebra.

Hindman’s theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. A Ramsey algebra is a structure that satisfies an analogue of Hindman’s theorem. In this paper, we present the basic notions of Ramsey algebras by using terminology from mathematical logic. We also present some results regarding classification of Ramsey algebras.

Hindman’s Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman’s Theorem. It is an open problem posed by Carlson whether every Ramsey algebra has an idempotent ultrafilter. This paper develops a general framework to study idempotent ultrafilters. Under certain countable setting, the main result roughly says that every nondegenerate Ramsey algebra has a nonprincipal idempotent ultrafilter in some nontrivial countable field of sets. This amounts to a positive result that addresses Carlson’s question in some way.

Given subset E of natural numbers FS(E) is defined as the collection of all sums of elements of finite subsets of E and any translation of FS(E) is said to be Hilbert cube. We can define the multiplicative analog of Hilbert cube as well. E.G. Strauss proved that for every @e>0 there exists a sequence with density >1-@e which does not contain an infinite Hilbert cube. On the other hand, Nathanson showed that any set of density 1 contains an infinite Hilbert cube. In the present note we estimate the density of Hilbert cubes which can be found avoiding sufficiently sparse (in particular, zero density) sequences. As a consequence we derive a result in which we ensure a dense additive Hilbert cube which avoids a multiplicative one.

Generalisations du theoreme de Gahvin-Prikry aux grands cardinaux. Etude de quelques proprietes de Ramsey duales pour les cardinaux non denombrables

For f ∈ `∞(N) let Tf be defined by Tf(n) = 1 n ∑n i=1 f(i). We investigate permutations g of N, which satisfy Tf(n)−Tfg(n)→ 0 as n→∞ with fg(n) = f(gn) for f ∈ `∞(N) (i.e. g is in the Lévy group G), or for f in the subspace of Cesàro-summable sequences. Our main interest are G-invariant means on `∞(N) or equivalently G-invariant probability measures on βN. We show that the adjoint T ∗ of T maps measures supported in βN \ N onto a weak*-dense subset of the space of G-invariant measures. We investigate the dynamical system (G, βN) and show that the support set of invariant measures on βN is the closure of the set of almost periodic points and the set of nontopologically transitive points in βN \ N. Finally we consider measures which are invariant under T ∗. Introduction For A ⊂ N the density function is given by dA(n) = 1 n |{i ∈ A : i ≤ n}|, and its upper density dA as dA = limn→∞ dA(n). A set A has density α if limn→∞ dA(n) = α. For a function f ∈ `∞(N) (the space of bounded real functions on N) we define the operator T by Tf(n) = 1 n ∑n i=1 f(i). Any element f of `∞(N) has a unique continuous extension to the Stone-Čech compactification βN of N. The continuous extension of the characteristic functions χA of a subset A of N is the characteristic function χA of the closure A of A in βN. For a general f in `∞(N) we will denote the continuous extension of a function f to βN again by f . This isometry between `∞(N) and C(βN) (the space of real continuous functions on βN) allows us to express continuous functions on βN by their values on the integers, i.e. as bounded sequences on N, to interpret T as a continuous operator on C(βN), and to identify the dual spaces of `∞(N) (finitely additive measures on N) and of C(βN) (regular Borel measures on βN). In particular means on `∞(N) correspond to probability measures on βN (see proof of Lemma 1 below). This correspondence between finitely additive measures on N and measures on βN becomes more transparent when we realize βN as the set of ultrafilters on N, with ultrafilters being finitely additive 0-1 valued measures (cf. [11], 20.37 (b)): Then for an ultrafilter q the Dirac measure δq is the 0-1 valued Borel measure on βN corresponding to the finitely additive 0-1 valued measure q on N. For our approach however it will be more convenient to work with the usual topological definition of a filter on a set M as a nonempty collection F of subsets of Received by the editors September 29, 1995. 1991 Mathematics Subject Classification. Primary 54H20. Part of this work was carried out at Macquarie University with financial support from the Australian Research Council. c ©1996 American Mathematical Society 5087 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5088 MARTIN BLÜMLINGER M satisfying (i) ∅ / ∈ F, (ii) A ∈ F, B ∈ F ⇒ A∩B ∈ F, (iii) A ∈ F, A ⊂ B ⊂M ⇒ B ∈ F. An ultrafilter is a filter with the additional property that it is not properly contained in any other filter (i.e. it is maximal), or equivalently, A ⊂M ⇒ A ∈ F or {A ∈ F. We identify the natural numbers with their neighborhood filters, so N ⊂ βN. A base for the topology on βN is now given by the clopen sets. A subset of βN is clopen iff it is the closure A of a subset A of N, i.e. iff it is the set of all ultrafilters containing A. With this abuse of notation we can see A ⊂ N both as a subset of βN and as an element of an ultrafilter p in A, i.e. we have p ∈ A iff A ∈ p. See [20] for a detailed exposition. A permutation g of N acts naturally as an automorphism on βN: For p ∈ βN let the ultrafilter gp be defined as gp = {gA : A ∈ p}. So gA ∈ gp ⇔ A ∈ p and g1(g2p) = (g1g2)p, i.e. the group of permutations of N acts as a transformation group on βN. Then for any clopen set A, A ⊂ N, we have gA = {gp : p ∈ A} = {gp : A ∈ p} = {gp : gA ∈ gp} = {q : gA ∈ q} = gA. (1) Since the clopen sets are a base for the topology on βN, it follows that g is an automorphism. A permutation g of N also induces an isometry of `∞(N) : f → fg, fg(n) = f(gn) and an isometry of C(βN) : f → fg, fg(p) = f(gp), p ∈ βN, so g acts naturally on the spaceM of (signed real regular Borel) measures on βN by 〈gμ, f〉 = 〈μ, fg〉 or equivalently by gμ(U) = μ(g−1U) for Borel sets U in βN. A measure μ in M is G-invariant (G being some group of permutations) if μ = μg ∀g ∈ G, i.e. if μ is in the annihilator {f − fg : f ∈ C(βN), g ∈ G}⊥ (with respect to the duality 〈M, C(βN)〉). We shall be interested in measures on βN, which are invariant under certain permutations defined via invariance of density functions at infinity. The Lévy group G is the group of permutations g satisfying Tf − Tfg vanishes on N∗ ∀f ∈ C(βN). We construct a weak*-dense subspace of the space of G-invariant measures and show that the bigger group Gδ of permutations satisfying this requirement only for Cesáro-summable bounded sequences f allows no Gδ-invariant measures. We investigate the topological dynamics of the permutation group G acting on βN; similar problems of invariance under right translations (or more generally under motions, i.e. 1-1 maps of N without periodic points) were discussed in [4], [12], [16] and [17]. See [2], II.5 for an overview. Although right translation τ of N is not a permutation of N, there are elements g in G which satisfy gn = τn except for a set of density 0 (cf. proof of Theorem 7). Since sets of density 0 will turn out to be negligible for the questions considered, it is safe to think of translations as being a subsemigroup of the Lévy group. In particular the action of τ on the support set S of measures invariant under the Lévy group coincides with the action of certain elements of the Lévy group on S. While the characterization of the support set Sτ of τ -invariant measures looks like our characterization of S (Theorem 4) with the notion of upper density replaced by that of upper Banach density, the orbit structures of the pertinent dynamical systems differ drastically: While we will show that points with minimal G-orbit closure are dense in S, it is known that there are open sets in Sτ without points with minimal τ -orbit closure. On the other hand every minimal τ -orbit closure is the support set of some τ -invariant measure, whereas there are minimal G-orbit closures which do not support any G-invariant measure. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use LÉVY GROUP ACTION AND INVARIANT MEASURES ON βN 5089 Some of these differences are due to the fact that the Lévy group is not amenable, whereas the semigroup of translations is amenable. An essential difference in the investigations of the dynamical systems (G, βN) and (τ, βN) stems from the fact that translations of N suggest the consideration of N as a semigroup under addition, which then allows some characterizations via algebraic properties of the left topological (right topological in the notation of [2]) semigroup βN. So the set S is a right ideal of the semigroup (βN,+) cf. [5], 7.4 and minimal closed right ideals are just minimal orbit closures under τ cf. [1], 2.1. Finally we characterize the support set of measures which are invariant under the adjoint T ∗ of T . This invariance is a stronger requirement than G-invariance. The support set SI of these measures allows a characterization similar to that of S or of SI , again with an appropriate notion of density. Permutation groups and invariant means on N We first clarify why the existence of positive invariant functionals is equivalent to the existence of nonzero invariant functionals for `∞(N) resp. C(βN). The proof of these facts is included for the reader’s convenience. We recall that a mean on N is a positive normalized finitely additive measure (set function) on N or equivalently a positive normalized functional on `∞(N). Lemma 1. For a group G of permutations of N the following are equivalent: (a) There exists a G-invariant mean on N; (b) There exists a nonzero G-invariant finitely additive measure on βN; (c) There exists a G-invariant probability measure on βN; (d) There exists a nonzero G-invariant (finite Borel) measure on βN. Proof. The restriction map from βN to N induces an isometry Φ between the Banach spaces C(βN) and `∞(N). The adjoint Φ∗ then is an isometry between their dual spaces: Φ∗ : `∞(N)∗ → C(βN)∗. The dual of `∞(N) is the space of finitely additive measures on N (cf. [6], IV 8.16) and the dual of C(βN) is the space M of (finite real regular Borel) measures on βN by the Riesz representation theorem (cf. [6], IV 6.2), both spaces with the total variation norm. We further note that, since for f ∈ C(βN) and a permutation g of N the action of g on f is defined by fg(p) = f(gp)∀p ∈ βN, we have (Φf)g(n) = (Φf)(gn) = f(gn) = fg(n) = Φ(fg)(n), (2) i.e. the action of G commutes with the isometry Φ and we may use the notation fg both for the continuous extension of fg ∈ `∞(N) to βN and for the action of g on the continuous extension of f ∈ `∞(N) to βN. Since the space of G-invariant measures on βN is just the annihilator of the set {f−fg : f ∈ C(βN), g ∈ G} and the space of G-invariant finitely additive measures on N is just the annihilator of the set {f − fg : f ∈ `∞(N), g ∈ G}, it follows that G-invariant measures on βN correspond to G-invariant finitely additive measures on N via the isometry Φ∗, since by (2) Φ is a bijection between the above two sets. Finally Φ is order preserving since the restriction to N of a positive function on C(βN) is positive, so means correspond to probability measures and (a) ⇔ (c) and (b) ⇔ (d) follow. Since (c)⇒ (d) is trivial, it remains to show (d) ⇒ (c): Let μ be a nontrivial Ginvariant measure on βN and let βN = X1∪X2 be the Hahn decomposition (cf. [11]) of βN, i.e. μ is nonnegative when restricted to X1 and nonpositive when restricted License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5090 MARTIN BLÜMLINGER to X2, wher