Hales-Jewett Theorem Research Articles

Carlson-Simpson's lemma and applications in reverse mathematics

We study the reverse mathematics of infinitary extensions of the Hales-Jewett theorem, due to Carlson and Simpson. These theorems have multiple applications in Ramsey's theory, such as the existence of finite big Ramsey degrees for the triangle-free graph, or the Dual Ramsey theorem. We show in particular that the Open Dual Ramsey theorem holds in ACA0+.

Hales–Jewett type configurations in small sets

In a recent work, N. Hindman, D. Strauss and L. Zamboni have shown that the Hales–Jewett theorem can be combined with a sufficiently well behaved homomorphisms. Their work was completely algebraic in nature, where they used the algebra of Stone–Čech compactification of discrete semigroups. They proved the existence of those configurations in piecewise syndetic sets, which are Ramsey theoretic rich sets. In our work we will show those forms are still present in very small but Ramsey theoretic sets, (like J-sets, C-sets) and our proof is purely elementary in nature.

Another Note on Intervals in the Hales–Jewett Theorem

The Hales–Jewett Theorem states that any $r$–colouring of $[m]^n$ contains a monochromatic combinatorial line if $n$ is large enough. Shelah's proof of the theorem implies that for $m = 3$ there always exists a monochromatic combinatorial line whose set of active coordinates is the union of at most $r$ intervals. For odd $r$, Conlon and Kamčev constructed $r$–colourings for which it cannot be fewer than $r$ intervals. However, we show that for even $r$ and large $n$, any $r$–colouring of $[3]^n$ contains a monochromatic combinatorial line whose set of active coordinates is the union of at most $r-1$ intervals. This is optimal and extends a result of Leader and Räty for $r=2$.

Topological Ramsey spaces of equivalence relations and a dual Ramsey theorem for countable ordinals

We define a collection of topological Ramsey spaces consisting of equivalence relations on ω with the property that the minimal representatives of the equivalence classes alternate according to a fixed partition of ω. To prove the associated pigeonhole principles, we make use of the left-variable Hales-Jewett theorem and its extension to an infinite alphabet. We also show how to transfer the corresponding infinite-dimensional Ramsey results to equivalence relations on countable limit ordinals (up to a necessary restriction on the set of minimal representatives of the equivalence classes) in order to obtain a dual Ramsey theorem for such ordinals.

Parametrized Ramsey theory of infinite block sequences of vectors

We show that the infinite-dimensional versions of Gowers' FINk and FIN±k theorems can be parametrized by an infinite sequence of perfect subsets of 2ω. To do so, we use ultra-Ramsey theory to obtain exact and approximate versions of a result which combines elements from both Gowers' theorems and the Hales-Jewett theorem. As a consequence, we obtain a parametrized version of Gowers' c0 theorem.

RAMSEY’S COHEIRS

AbstractWe use the model theoretic notion of coheir to give short proofs of old and new theorems in Ramsey Theory. As an illustration we start from Ramsey’s theorem itself. Then we prove Hindman’s theorem and the Hales–Jewett theorem. Finally, we prove two Ramsey theoretic principles that have among their consequences partition theorems due to Carlson and to Gowers.

A structure theorem for stochastic processes indexed by the discrete hypercube

Abstract Let A be a finite set with , let n be a positive integer, and let $A^n$ denote the discrete $n\text {-dimensional}$ hypercube (that is, $A^n$ is the Cartesian product of n many copies of A). Given a family $\langle D_t:t\in A^n\rangle $ of measurable events in a probability space (a stochastic process), what structural information can be obtained assuming that the events $\langle D_t:t\in A^n\rangle $ are not behaving as if they were independent? We obtain an answer to this problem (in a strong quantitative sense) subject to a mild ‘stationarity’ condition. Our result has a number of combinatorial consequences, including a new (and the most informative so far) proof of the density Hales-Jewett theorem.

Applications of the Hales–Jewett theorem near zero

The famous van der Waerden’s theorem states that if $${\mathbb{N}}$$ is finitely colored then one color class will contain arithmetic progressions of arbitrary length. The polynomial van der Waerden’s theorem says that if $$p_{1}(x),p_2(x),\ldots ,p_{k}(x)$$ are polynomials with integer coefficients and zero constant term and $${\mathbb{N}}$$ is finitely colored, then there exist $$a,d\in {\mathbb{N}}$$ such that $$\big \{a+p_{t}(d):t\in \{1,2,\ldots ,k\} \big \}$$ is monochromatic. There are dynamical, algebraic, and combinatorial proofs of this theorem. In this article we will prove a “near zero” version of the polynomial van der Waerden’s theorem. That is, we will show that for any $$\epsilon >0$$ , if $$(0,\epsilon )\cap {\mathbb{Q}}$$ is finitely colored then there exists $$a,d\in {\mathbb{Q}}{\setminus }\{0\}$$ such that $$\big \{a+p_{t}(d):t\in \{0,1,2,\ldots ,k\}\big \}$$ is monochromatic.

Combining Extensions of the Hales-Jewett Theorem with Ramsey Theory in Other Structures

The Hales-Jewett Theorem states that given any finite nonempty set ${\mathbb A}$ and any finite coloring of the free semigroup $S$ over the alphabet ${\mathbb A}$ there is a variable word over ${\mathbb A}$ all of whose instances are the same color. This theorem has some extensions involving several distinct variables occurring in the variable word. We show that, when combined with a sufficiently well behaved homomorphism, the relevant variable word simultaneously satisfies a Ramsey-Theoretic conclusion in the other structure. As an example we show that if $\tau$ is the homomorphism from the set of variable words into the natural numbers which associates to each variable word $w$ the number of occurrences of the variable in $w$, then given any finite coloring of $S$ and any infinite sequence of natural numbers, there is a variable word $w$ whose instances are monochromatic and $\tau(w)$ is a sum of distinct members of the given sequence.Our methods rely on the algebraic structure of the Stone-Čech compactification of $S$ and the other semigroups that we consider. We show for example that if $\tau$ is as in the paragraph above, there is a compact subsemigroup $P$ of $\beta{\mathbb N}$ which contains all of the idempotents of $\beta{\mathbb N}$ such that, given any $p\in P$, any $A\in p$, and any finite coloring of $S$, there is a variable word $w$ whose instances are monochromatic and $\tau(w)\in A$.We end with a new short algebraic proof of an infinitary extension of the Graham-Rothschild Parameter Sets Theorem.

On the number of points in general position in the plane

On the number of points in general position in the plane, Discrete Analysis 2018:16, 20 pp. A recurring theme in combinatorics is questions of the following kind. Suppose that we have a combinatorial structure $S$ of size $n$ that contains no object of type $A$. Then how large a subset of $S$ can we find that contains no object of type $B$? For example, a graph with $n$ vertices that contains no clique of size 4 can be shown quite easily to have a triangle-free subgraph with $n^{1/2}$ vertices, and Wolfovitz has shown that there are graphs with no clique of size 4 and no triangle-free subgraph with more than $n^{1/2}(\log n)^{120}$ vertices. One of the main questions discussed in this paper is perhaps the first question of this type one would think of in discrete geometry: if $S$ is a set of $n$ points in the plane and if no four of these points are collinear, then how large a subset of $S$ can one find with no three collinear? A bound of $o(n)$ follows from the density Hales-Jewett theorem, which implies that a subset of $\{1,2,3\}^k$ of positive density contains three points in a line. It is not hard to project the set $\{1,2,3\}$ into the plane in such a way that collinearity is preserved, but no four points of the image lie in a line. However, the bound obtained this way is very weak -- roughly $n/\log_*(n)$. This paper obtains the first reasonable bound for the problem, namely $n^{5/6+o(1)}$. It is not clear whether 5/6 is the right exponent, but the authors suggest that their construction may be close to optimal and that the difficulty is to calculate the correct exponent for that example. Perhaps the most interesting aspect of the paper is that it uses the so-called method of containers. This method, developed by Saxton and Thomason, and independently by Balogh, Morris and Samotij, has already been used to solve a large number of important problems, but this appears to be the first time it has been used to solve a problem in discrete geometry, and it is used in a novel way. They also use containers to prove a second discrete geometry result, this time about epsilon-nets. Given a family $\mathcal F$ of subsets of a finite set $X$, an $\epsilon$-net $E$ of $\mathcal F$ is a subset $E$ of $X$ such that every $F\in\mathcal F$ of size at least $\epsilon|X|$ contains an element of $E$. There are many interesting questions about the sizes of $\epsilon$-nets when $X$ is a geometrical set such as a finite set of points in the plane, and $\mathcal F$ is some natural class of subsets such as the set of all intersections of $X$ with convex bodies. With this example, one can also define a _weak_ $\epsilon$-net as follows: it is a set of points $E$ in the plane, not necessarily a subset of $X$, such that every convex hull of at least $\epsilon|X|$ points of $X$ contains a point of $E$. Natural notions of weak $\epsilon$-nets can be defined in many other contexts too. An interesting open question, asked by Noga Alon, is whether there is some natural geometrically defined family $\mathbb F$ of bounded VC-dimension such that the smallest $\epsilon$-net has size at least $(c/\epsilon)\log(1/\epsilon)$. Also using the density Hales-Jewett theorem, Alon obtained a bound that was very slightly superlinear in the case where $X$ was a certain point set and $\mathcal F$ was the set of all intersections of lines with $X$. In this paper, Alon's bound is improved to $(1/\epsilon)\log(1/\epsilon)^{1/3-o(1)}$, which is much closer to the bound he suggests might be obtainable. They also obtain an improved bound for weak $\epsilon$-nets, but with a power of $\log\log(1/\epsilon)$ replacing the power of $\log(1/\epsilon)$. This construction has the additional feature that it works just as well in the projective plane.

A Note on Intervals in the Hales-Jewett Theorem

The Hales-Jewett theorem for alphabet of size 3 states that whenever the Hales-Jewett cube $[3]^{n}$ is $r$-coloured there is a monochromatic line (for n large). Conlon and Kamcev conjectured that, for any $n$, there is a 2-colouring of $[3]^{n}$ for which there is no monochromatic line whose active coordinate set is an interval. In this note we disprove this conjecture.

A note on multiparty communication complexity and the Hales–Jewett theorem

For integers n and k, the density Hales–Jewett numbercn,k is defined as the maximal size of a subset of [k]n that contains no combinatorial line. We show that for k≥3 the density Hales–Jewett number cn,k is equal to the maximal size of a cylinder intersection in the problem Partn,k of testing whether k subsets of [n] form a partition. It follows that the communication complexity, in the Number On the Forehead (NOF) model, of Partn,k, is equal to the minimal size of a partition of [k]n into subsets that do not contain a combinatorial line. Thus, the bound in [7] on Partn,k using the Hales–Jewett theorem is in fact tight, and the density Hales–Jewett number can be thought of as a quantity in communication complexity. This gives a new angle to this well studied quantity.As a simple application we prove a lower bound on cn,k, similar to the lower bound in [19] which is roughly cn,k/kn≥exp⁡(−O(log⁡n)1/⌈log2⁡k⌉). This lower bound follows from a protocol for Partn,k. It is interesting to better understand the communication complexity of Partn,k as this will also lead to the better understanding of the Hales–Jewett number. The main purpose of this note is to motivate this study.

Revisiting the nilpotent polynomial Hales–Jewett theorem

Answering a question posed by Bergelson and Leibman in [6], we establish a nilpotent version of the Polynomial Hales–Jewett Theorem that contains the main theorem in [6] as a special case. Important to the formulation and the proof of our main theorem is the notion of a relative syndetic set (relative with respect to a closed non-empty subsets of βG) [25]. As a corollary of our main theorem we prove an extension of the restricted van der Waerden Theorem to nilpotent groups, which involves nilprogressions.

A short proof of the induced Ramsey Theorem for hypergraphs

For fixed r≥2, an r-graph G=(V,E) is an r-uniform hypergraph with vertex set V and edge set E⊆(Vr). For k≥r≥2, let Kk denote the complete r-graph on k vertices and let Kk¯ denote its complement, an independent set on k vertices.The Induced Ramsey Theorem states that for c,r≥2 and every r-graph G, there exists an r-graph H such that every c-coloring of the edges of H contains a monochromatic induced copy of G. A natural question to ask is what other subgraphs F can be partitioned and have a similar Ramsey property. One can show that if F≠Kk or F≠Kk¯, then this fails to be true. On the other hand, a result of Abramson and Harrington (1978) and Nešetřil and Rödl (1977) implies that F=Kk, as well as F=Kk¯, has the Ramsey property.The proof of the Induced Ramsey Theorem was based on a partite lemma and partite construction as shown in Nešetřil and Rödl (1977) and Nešetřil and Rödl (1987). In this note, we present a short proof of this result by eliminating the partite construction to show that the theorem is a direct consequence of the Hales–Jewett Theorem. In other words, we have reduced the proof of this result to one step, rather than two steps.

A density Hales–Jewett theorem for matroids

We show that if α is a positive real number, n and ℓ are integers exceeding 1, and q is a prime power, then every simple matroid M of sufficiently large rank, with no U2,ℓ-minor, no rank-n projective geometry minor over a larger field than GF(q), and at least αqr(M) elements, has a rank-n affine geometry restriction over GF(q). This result can be viewed as an analogue of the multidimensional density Hales–Jewett theorem for matroids.

On the Grid Ramsey Problem and Related Questions: Fig. 1.

The Hales–Jewett theorem is one of the pillars of Ramsey theory, from which many other results follow. A celebrated theorem of Shelah says that Hales–Jewett numbers are primitive recursive. A key tool used in his proof, now known as the cube lemma, has become famous in its own right. In its simplest form, this lemma says that if we color the edges of the Cartesian product Kn × Knin r colors, then, for nsufficiently large, there is a rectangle with both pairs of opposite edges receiving the same color. Shelah’s proof shows that n=r � r+1 2 � + 1 suffices. More than 20 years ago, Graham, Rothschild, and Spencer asked whether this bound can be improved to a polynomial in r. We show that this is not possible by providing a superpolynomial lower bound in r. We also discuss a number of related problems.

An analogue of the Erdős-Stone theorem for finite geometries

For a set G of points in PG(m?1,q), let exq(G;n) denote the maximum size of a collection of points in PG(n?1,q) not containing a copy of G, up to projective equivalence. We show that [Figure not available: see fulltext.] where c is the smallest integer such that there is a rank-(m?c) flat in PG(m?1,q) that is disjoint from G. The result is an elementary application of the density version of the Hales-Jewett Theorem.

Ramsey theory for discrete structures

This monograph covers some of the most important developments in Ramsey theory from its beginnings in the early 20th century via its many breakthroughs to recent important developments in the early 21st century. The book first presents a detailed discussion of the roots of Ramsey theory before offering a thorough discussion of the role of parameter sets. It presents several examples of structures that can be interpreted in terms of parameter sets and features the most fundamental Ramsey-type results for parameter sets: Hales-Jewett's theorem and Graham-Rothschilds Ramsey theorem as well as their canonical versions and several applications. Next, the book steps back to the most basic structure, to sets. It reviews classic results as well as recent progress on Ramsey numbers and the asymptotic behavior of classical Ramsey functions. In addition, it presents product versions of Ramsey's theorem, a combinatorial proof of the incompleteness of Peano arithmetic, provides a digression to discrepancy theory and examines extensions of Ramsey's theorem to larger cardinals. The next part of the book features an in-depth treatment of the Ramsey problem for graphs and hypergraphs. It gives an account on the existence of sparse and restricted Ramsey theorem's using sophisticated constructions as well as probabilistic methods. Among others it contains a proof of the induced Graham-Rothschild theorem and the random Ramsey theorem. The book closes with a chapter on one of the recent highlights of Ramsey theory: a combinatorial proof of the density Hales-Jewett theorem. This book provides graduate students as well as advanced researchers with a solid introduction and reference to the field.

A nilpotent IP polynomial multiple recurrence theorem

We generalize the IP-polynomial Szemerédi theorem due to Bergelson and McCutcheon and the nilpotent Szemerédi theorem due to Leibman. Important tools in our proof include a generalization of Leibman’s result that polynomial mappings into a nilpotent group form a group and a multiparameter version of the nilpotent Hales-Jewett theorem due to Bergelson and Leibman.

Abstract approach to finite Ramsey theory and a self-dual Ramsey theorem

We give an abstract approach to finite Ramsey theory and prove a general Ramsey-type theorem. We deduce from it a self-dual Ramsey theorem, which is a new result naturally generalizing both the classical Ramsey theorem and the dual Ramsey theorem of Graham and Rothschild. In fact, we recover the pure finite Ramsey theory from our general Ramsey-type result in the sense that the classical Ramsey theorem, the Hales–Jewett theorem (with Shelahʼs bounds), the Graham–Rothschild theorem, the versions of these results for partial rigid surjections due to Voigt, and the new self-dual Ramsey theorem are all obtained as iterative applications of the general result.