Van Der Waerden's Theorem Research Articles

An infinitary polynomial Hales-Jewett theorem

A joint extension of H. Furstenberg’s central sets theorem, the Hales-Jewett coloring theorem and the polynomial van der Waerden theorem of V. Bergelson and A. Leibman is obtained by an elaboration on Furstenberg and Y. Katznelson’s approach to infinitary Ramsey theory via the enveloping semigroup.

Combinatorial Proofs of the Polynomial van der Waerden Theorem and the Polynomial Hales-Jewett Theorem

Recently Bergelson and Leibman proved an extension to van der Waerden's theorem involving polynomials. They also generalised the Hales–Jewett theorem in a similar way. The paper presents short and purely combinatorial proofs of those results.

Set-Polynomials and Polynomial Extension of the Hales-Jewett Theorem

An abstract, Hales-Jewett type extension of the polynomial van der Waerden Theorem [BL] is established: Theorem. Let r, d, q ∈ N. There exists N ∈ N such that for any r-coloring of the set of subsets of V = {1, . . . , N}d×{1, . . . , q} there exist a set a ⊂ V and a nonempty set γ ⊆ {1, . . . , N} such that a∩ (γd × {1, . . . , q}) = ∅, and the subsets a, a ∪ (γd × {1}), a ∪ (γd × {2}), . . ., a ∪ (γd × {q}) are all of the same color. This “polynomial” Hales-Jewett theorem contains refinements of many combinatorial facts as special cases. The proof is achieved by introducing and developing the apparatus of set-polynomials (polynomials whose coefficients are finite sets) and applying the methods of topological dynamics.

An Infinitary Polynomial van der Waerden Theorem

In the spirit of Hindman's theorem, we prove a joint extension of an infinitary van der Waerden theorem of Furstenberg and a polynomial van der Waerden theorem of Bergelson and Leibman.

A van der Waerden Variant

The classical van der Waerden Theorem says that for every every finite set $S$ of natural numbers and every $k$-coloring of the natural numbers, there is a monochromatic set of the form $aS+b$ for some $a>0$ and $b\geq 0$. I.e., monochromatism is obtained by a dilation followed by a translation. We investigate the effect of reversing the order of dilation and translation. $S$ has the variant van der Waerden property for $k$ colors if for every $k$-coloring there is a monochromatic set of the form $a(S+b)$ for some $a>0$ and $b\geq 0$. On the positive side it is shown that every two-element set has the variant van der Waerden property for every $k$. Also, for every finite $S$ and $k$ there is an $n$ such that $nS$ has the variant van der Waerden property for $k$ colors. This extends the classical van der Waerden Theorem. On the negative side it is shown that if $S$ has at least three elements, the variant van der Waerden property fails for a sufficiently large $k$. The counterexamples to the variant van der Waerden property are constructed by specifying colorings as Thue-Morse sequences.

Recurrent Ideas in Number Theory: The Multiple Birkhoff Recurrence Theorem Used to Prove van der Waerden's Theorem

In 1977 and 1978, some striking and beautiful connections between the fields of topological dynamics and combinatorial number theory began to emerge. Since then there has been an explosion of sensational results that connect these two fields and provide new and clear insights into some classical number-theoretic theorems and combinatorial questions. This story is about classical number-theoretic results being understood in a new and systematic way using the tools and machinery of dynamical systems. The point of this note is to discuss in some detail one such important number-theoretic result and its new dynamical interpretation, namely van der Waerden's theorem, which provides a particularly clear and simple example. (We focus on the dynamical interpretation of this number-theoretic result. Classical proofs of theorems are not included, but the reader is invited to explore the references [6], [71, [2], [8].) Number theory is one of the oldest branches of mathematics. It deals with the natural numbers 1N, which are used for counting and calculations, and it is the solid foundation of modern algebra. Combinatorial number theory, simply speaking, involves questions about sets of natural numbers and classes of sets of natural numbers and the structure of these classes as sets. Questions in combinatorial number theory are quantitative and set-theoretic in nature, as opposed to those concerning algebraic properties of numbers (divisibility, primeness, etc.). In the late 1920's the following problem was posed: If the natural numbers are divided into 2 classes, must there be an arithmetic progression of arbitrary length in one of the classes? Many mathematicians tried to solve this problem-it is appealing due to its (deceptively) simple nature -but the solution by van der Waerden proved to be surprisingly deep. In any event, in 1927 van der Waerden proved the following theorem, answering a slightly more general question.

Minimal invariant spaces in formal topology

A standard result in topological dynamics is the existence of minimal subsystem. It is a direct consequence of Zorn's lemma: given a compact topological space X with a map f: X→X, the set of compact non empty subspaces K of X such that f(K) ⊆ K ordered by inclusion is inductive, and hence has minimal elements. It is natural to ask for a point-free (or formal) formulation of this statement. In a previous work [3], we gave such a formulation for a quite special instance of this statement, which is used in proving a purely combinatorial theorem (van de Waerden's theorem on arithmetical progression).In this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a point-free formulation of the existence of a minimal subspace for any continuous map f: X→X. We show that such minimal subspaces can be described as points of a suitable formal topology, and the “existence” of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is “similar in structure” to the topological proof [6, 8], but which uses a simple algebraic remark (Proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements.

Elementary proof for a Van der Waerden's conjecture and related theorems

A well-known conjecture of Van der Waerden says that for the permanent Per A of an n × n doubly stochastic matrix A we have (3.0), with equality if and only if all entries of the matrix A equal to n −1 . In 1977 [1], the author proved that if A is an n × n doubly stochastic matrix, and p ≥ 0, q ≥ 0, p + q = 1, then (2.0) holds with equality if and only if all entries of the matrix A equal to n −1 . In this paper, we show that (3.0) and (2.0) are equivalent. On the basis of this equivalence one can say that the equivalent of the Van der Waerden's conjecture was solved already in 1977. A further subject of the paper is to show similar equivalence theorems concerning permanents of doubly stochastic matrices, moreover a refinement of the Van der Waerden's theorem. A separate section deals with the probabilistic interpretation of some previous results.

A constructive topological proof of van der Waerden's theorem

The theorem of van der Waerden on arithmetical progression is a classic result in combinatorics. While its original proof was of combinatorial nature, it was shown by Furstenberg and Weiss that this theorem can be derived from topological dynamics. This last derivation is non-effective, and it is an interesting proof-theoretical problem to extract the computational content of this topological proof. This has been done by Girard using Kreisel's no counterexample interpretation. Here, we give a direct constructive formulation of this topological proof, using basic notions of point-free topology.

Generalized quantifiers and pebble games on finite structures

First-order logic is known to have a severely limited expressive power on finite structures. As a result, several different extensions have been investigated, including fragments of second-order logic, fixpoint logic, and the infinitary logic L ∞ ω ω in which every formula has only a finite number of variables. In this paper, we study generalized quantifiers in the realm of finite structures and combine them with the infinitary logic L ∞ ω ω to obtain the logics L ∞ ω ω ( Q), where Q = { Q i : iϵ I} is a family of generalized quantifiers on finite structures. Using the logics L ∞ ω ω ( Q), we can express polynomial-time properties that are not definable in L ∞ ω ω , such as “there is an even number of x” and “there exists at least n 2 x ” ( n is the size of the universe), without going to second-order logic. We show that equivalence of finite structures relative to L ∞ ω ω ( Q) can be characterized in terms of certain pebble games that are a variant of the Ehrenfeucht—Fraïssé games. We combine this game-theoretic characterization with sophisticated combinatorial tools from Ramsey theory, such as van der Waerden's Theorem and Folkman's Theorem, in order to investigate the scope and limits of generalized quantifiers in finite model theory. We obtain sharp lower bounds for expressibility in the logics L ∞ ω ω ( Q) and discover an intrinsic difference between adding finitely many simple unary generalized quantifiers to L ∞ ω ω adding infinitely many. In particular, we show that if Qis a finite sequence of simple unary generalized quantifiers, then the equicardinality, or Härtig, quantifier is not definable in L ∞ ω ω ( Q). We also show that the query “does the equivalence relation E have an even number of equivalence classes” is not definable in the extension L ∞ ω ω ( I, Q) of L ∞ ω ω by the Härtig quantifier I and any finite sequence Q of simple unary generalized quantifiers.

Coloring the edges of km × km

AbstractIn the graph Km × Km a 4‐cycle is called square if its vertices do not all lie in the same copy of Km. A c‐coloring of the edges of Km × Km is good if in any square 4‐cycle the two edges of at least one nonadjacent pair receive different colors. Let f(2,c) denote the smallest integer m so that no c‐coloring of Km × Km is good. We show that there exists a constant k so that f(2,c) > kc3. (This function was introduced by Shelah in his proof of van der Waerden's theorem).

Ultrafilters and multidimensional Ramsey Theorems

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.

Arithmetic progressions in partially ordered sets

Van der Waerden's arithmetic sequence theorem—in particular, the ‘density version’ of Szemeredi—is generalized to partially ordered sets in the following manner. Let w and t be fixed positive integers and e>0. Then for every sufficiently large partially ordered set P of width at most w, every subset S of P satisfying |S|≥e|P| contains a chain a1, a2,..., a1 such that the cardinality of the interval [ai, ai+1] in P is the same for each i.

Behrend's Theorem for Dense Subsets of Finite Vector Spaces

The “combinatorial line conjecture” states that for all q ≧ 2 and there exists such that if , X is a q-element set, and A is any subset of X n (= cartesian product of n copies of X) with more than elements (that is, A has density greater than ), then A contains a combinatorial line. (For a definition of combinatorial line, together with statements and proofs of many results related to the combinatorial line conjecture, including all those results mentioned below, see [5]. Since we are not directly concerned with combinatorial lines in this paper, we do not reproduce the definition here.)This conjecture (which is a strengthened version of a conjecture of Moser [7]), if true, would bear the same relation to the Hales-Jewett theorem that Szemerédi's theorem bears to van der Waerden's theorem.

A canonical partition theorem for equivalence relations on Zt

A canonical version of the multidimensional version of van der Waerden's theorem on arithmetic progressions is proved.

A quintessential proof of van der Waerden's theorem on arithmetic progressions

A particularly well suited induction hypothesis is employed to give a short and relatively direct formulation of van der Waerden's argument which establishes that for any partition of the natural numbers into two classes, one of the classes contains arbitrarily long arithmetic progressions.

A note on van der Waerden's theorem

A note on van der Waerden's theorem

On van der Waerden's theorem on arithmetic progressions

On van der Waerden's theorem on arithmetic progressions

On van der Waerden's theorem and the theorem of Paris and Harrington

A 2-coloring of the non-negative integers and a function h are given such that if P is any monochromatic arithmetic progression with first term a and common difference d then ‖ P‖ ⩽ h( a) and ‖ P‖ ⩽ h( d). In contrast to this the following result is noted. For any k, f there is n = n( k, f) such that whenever n is k-colored there is a monochromatic subset A of n with ‖ A‖ > f( d), where d is the maximum of the differences between consecutive elements of A.

Arithmetic Progressions Contained in Sequences with Bounded Gaps

Van der Waerden [1, 4, 5] proved that if the nonnegative integers are partitioned into a finite number of sets, then at least one set in the partition contains arbitrarily long finite arithmetic progressions. This is equivalent to the result that a strictly increasing sequence of integers with bounded gaps contains arbitrarily long finite arithmetic progressions. Szemerèdi [3] proved the much deeper result that a sequence of integers of positive density contains arbitrarily long finite arithmetic progressions. The purpose of this note is a quantitative comparison of van der Waerden's theorem and sequences with bounded gaps.