Finite Coloring Research Articles

Monochromatic products and sums in the rationals

Abstract We show that every finite coloring of the rationals contains monochromatic sets of the form $\{x,y,xy,x+y\}$ .

Monochromatic quotients, products and polynomial sums in the rationals

Let k,a∈N and let p1,⋯,pk∈Q[n] with zero constant term. We show that for any finite coloring of Q, there are non-zero x,y∈Q such that there exists a color which contains the set{x,xya,x+p1(y),⋯,x+pk(y)} and there are non-zero v,u∈Q such that there exists a color which contains the set{v,v⋅ua,v+p1(u),⋯,v+pk(u)}.

Monochromatic exponential triples: An ultrafilter proof

We present a short ultrafilter proof of the existence of monochromatic exponential triples { a , b , b a } \{a, b, b^a\} in any finite coloring of the natural numbers. We then generalize the construction and prove a new result on the existence of infinite monochromatic exponential patterns.

Decomposing Aronszajn lines

We show that under the proper forcing axiom the class of all Aronszajn lines behave like [Formula: see text]-scattered orders under the embeddability relation. In particular, we are able to show that the class of better-quasi-order labeled fragmented Aronszajn lines is itself a better-quasi-order. Moreover, we show that every better-quasi-order labeled Aronszajn line can be expressed as a finite sum of labeled types which are algebraically indecomposable. By encoding lines with finite labeled trees, we are also able to deduce a decomposition result, that for every Aronszajn line [Formula: see text] there is an integer [Formula: see text] such that for any finite coloring of [Formula: see text] there is subset [Formula: see text] of [Formula: see text] isomorphic to [Formula: see text] which uses no more than [Formula: see text] colors.

An asymmetric random Rado theorem for single equations: The 0‐statement

AbstractA famous result of Rado characterizes those integer matrices A which are partition regular, that is, for which any finite coloring of the positive integers gives rise to a monochromatic solution to the equation . Aigner‐Horev and Person recently stated a conjecture on the probability threshold for the binomial random set having the asymmetric random Rado property: given partition regular matrices (for a fixed ), however one r‐colors , there is always a color such that there is an i‐colored solution to . This generalizes the symmetric case, which was resolved by Rödl and Ruciński, and Friedgut, Rödl and Schacht. Aigner‐Horev and Person proved the 1‐statement of their asymmetric conjecture. In this paper, we resolve the 0‐statement in the case where the correspond to single linear equations. Additionally we close a gap in the original proof of the 0‐statement of the (symmetric) random Rado theorem.

Formula omitted] does not imply [formula omitted] in ω-models

We complete a 40-year old program on the computability-theoretic analysis of Ramsey's theorem, starting with Specker and Jockusch in 1971/1972, by improving a result of Chong, Slaman and Yang in 2014. Given a set X, let [X]n be the collection of all n-element subsets of X. Ramsey's theorem for n-tuples asserts the existence, for every finite coloring of [ω]n, of an infinite set X⊆ω such that [X]n is monochromatic. The meta-mathematical study of Ramsey theory has a rich history, with several long-standing open problems and seminal theorems, including Seetapun's theorem in 1995 and Liu's theorem in 2012 about Ramsey's theorem for pairs. The remaining question about the study of Ramsey's theorem from a computational viewpoint was the relation between Ramsey's theorem for pairs (RT22) and its restriction to stable colorings (SRT22), that is, colorings admitting a limit behavior. Chong, Slaman and Yang first proved that SRT22 does not formally imply RT22 in a proof-theoretic sense, using non-standard models of reverse mathematics. In this article, we answer the open question whether this non-implication also holds within the framework of computability theory. More precisely, we construct an ω-model of SRT22 which is not a model of RT22. For this, we design a new notion of effective forcing refining Mathias forcing using the notion of largeness classes.

On the strength of Ramsey's theorem for trees

Let TT1 denote the principle that every finite coloring of the full binary tree has a homogeneous isomorphic subtree, i.e. one that is monochromatic. We show that over the system RCA0, the inductive strength of TT1 is weaker than Σ20-induction. This follows from the main theorem that over the same system, TT1 is Π11-conservative over Σ20-bounding plus the principle of the totality of the Ackermann function.

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.

A TAIL CONE VERSION OF THE HALPERN–LÄUCHLI THEOREM AT A LARGE CARDINAL

AbstractThe classical Halpern–Läuchli theorem states that for any finite coloring of a finite product of finitely branching perfect trees of height ω, there exist strong subtrees sharing the same level set such that tuples in the product of the strong subtrees consisting of elements lying on the same level get the same color. Relative to large cardinals, we establish the consistency of a tail cone version of the Halpern–Läuchli theorem at a large cardinal (see Theorem 3.1), which, roughly speaking, deals with many colorings simultaneously and diagonally. Among other applications, we generalize a polarized partition relation on rational numbers due to Laver and Galvin to one on linear orders of larger saturation.

A computable analysis of variable words theorems

The Carlson-Simpson lemma is a combinatorial statement occurring in the proof of the Dual Ramsey theorem. Formulated in terms of variable words, it informally asserts that given any finite coloring of the strings, there is an infinite sequence with infinitely many variables such that for every valuation, some specific set of initial segments is homogeneous. Friedman, Simpson, and Montalban asked about its reverse mathematical strength. We study the computability-theoretic properties and the reverse mathematics of this statement, and relate it to the finite union theorem. In particular, we prove the Ordered Variable word for binary strings in ACA0.

A structure theorem for multiplicative functions over the Gaussian integers and applications

We prove a structure theorem for multiplicative functions on the Gaussian integers, showing that every bounded multiplicative function on the Gaussian integers can be decomposed into a term which is approximately periodic and another which has a small U3-Gowers uniformity norm. We apply this to prove partition regularity results over the Gaussian integers for certain equations involving quadratic forms in three variables. For example, we show that for any finite coloring of the Gaussian integers, there exist distinct nonzero elements x and y of the same color such that x2−y2 = n2 for some Gaussian integer n. The analog of this statement over Z remains open.

Algebra, selections, and additive Ramsey theory

Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods include, in addition to Hurewicz's game theoretic characterization of Menger's property, extensions of the classic idempotent theory in the Stone--Czech compactification of semigroups, and of the more recent theory of selection principles. This provides strong versions of the mentioned celebrated theorems, where the monochromatic substructures are large, beyond infinitude, in an analytic sense. Reducing the main theorems to the purely combinatorial setting, we obtain nontrivial consequences concerning uncountable cardinal characteristics of the continuum. The main results, modulo technical refinements, are of the following type (definitions provided in the main text): Let $X$ be a Menger space, and $\mathcal{U}$ be an infinite open cover of $X$. Consider the complete graph, whose vertices are the open sets in $X$. For each finite coloring of the vertices and edges of this graph, there are disjoint finite subsets $\mathcal{F}_1,\mathcal{F}_2,\dots$ of the cover $\mathcal{U}$ whose unions $V_1 := \bigcup\mathcal{F}_1, V_2 := \bigcup\mathcal{F}_2,\dots$ have the following properties: 1. The sets $\bigcup_{n\in F}V_n$ and $\bigcup_{n\in H}V_n$ are distinct for all nonempty finite sets $F<H$. 2. All vertices $\bigcup_{n\in F}V_n$, for nonempty finite sets $F$, are of the same color. 3. All edges $\bigl\{\,\bigcup_{n\in F}V_n, \bigcup_{n\in H}V_n\,\bigr\}$, for nonempty finite sets $F<H$, have the same color. 4. The family $\{V_1,V_2,\dots\}$ is an open cover of $X$. A self-contained introduction to the necessary parts of the needed theories is provided.

Coloring trees in reverse mathematics

The tree theorem for pairs (TT22), first introduced by Chubb, Hirst, and McNicholl, asserts that given a finite coloring of pairs of comparable nodes in the full binary tree 2<ω, there is a set of nodes isomorphic to 2<ω which is homogeneous for the coloring. This is a generalization of the more familiar Ramsey's theorem for pairs (RT22), which has been studied extensively in computability theory and reverse mathematics. We answer a longstanding open question about the strength of TT22, by showing that this principle does not imply the arithmetic comprehension axiom (ACA0) over the base system, recursive comprehension axiom (RCA0), of second-order arithmetic. Combined with a recent result of Patey's that TT22 is strictly stronger than RT22, this establishes TT22 as the first known example of a natural combinatorial principle to occupy the interval strictly between ACA0 and RT22. The proof of this fact uses an extension of the bushy tree forcing method, and develops new techniques for dealing with combinatorial statements formulated on trees, rather than on ω.

Monochromatic sums and products in $\mathbb{N}$

An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair $\{x+y,xy\}$. We answer this question affirmatively in a strong sense by exhibiting a large new class of non-linear patterns which can be found in a single cell of any finite partition of $\mathbb{N}$. Our proof involves a correspondence principle which transfers the problem into the language of topological dynamics. As a corollary of our main theorem we obtain partition regularity for new types of equations, such as $x^2-y^2=z$ and $x^2+2y^2-3z^2=w$.

Are Monochromatic Pythagorean Triples Unavoidable under Morphic Colorings?

ABSTRACTA Pythagorean triple is a triple of positive integers satisfying a2 + b2 = c2. Is it true that for any finite coloring of , at least one Pythagorean triple must be monochromatic? In other words, is the Diophantine equation X2 + Y2 = Z2 regular? This problem, recently solved for 2-colorings by massive SAT computations [Heule et al., 2016], remains widely open for k-colorings with k ⩾ 3. In this article, we introduce morphic colorings of , which are special colorings in finite groups with partly multiplicative properties. We show that for many morphic colorings in 2 and 3 colors, monochromatic Pythagorean triples are unavoidable in rather small integer intervals.

Ramsey Algebras and Formal Orderly Terms

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.

Ramsey algebras

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. This paper introduces Ramsey algebras and presents some elementary results. Furthermore, their connection to Ramsey spaces will be addressed.

Monochromatic paths for the integers

Recall that van der Waerden’s theorem states that any finite coloring of the naturals has arbitrarily long monochromatic arithmetic sequences. We explore questions about the set of differences of those sequences.

Ramsey algebras and the existence of idempotent ultrafilters

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 developes 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.

Ergodic theorem involving additive and multiplicative groups of a field and patterns

We establish a ‘diagonal’ ergodic theorem involving the additive and multiplicative groups of a countable field $K$ and, with the help of a new variant of Furstenberg’s correspondence principle, prove that any ‘large’ set in $K$ contains many configurations of the form $\{x+y,xy\}$. We also show that for any finite coloring of $K$ there are many $x,y\in K$ such that $x,x+y$ and $xy$ have the same color. Finally, by utilizing a finitistic version of our main ergodic theorem, we obtain combinatorial results pertaining to finite fields. In particular, we obtain an alternative proof for a result obtained by Cilleruelo [Combinatorial problems in finite fields and Sidon sets. Combinatorica32(5) (2012), 497–511], showing that for any finite field $F$ and any subsets $E_{1},E_{2}\subset F$ with $|E_{1}|\,|E_{2}|&gt;6|F|$, there exist $u,v\in F$ such that $u+v\in E_{1}$ and $uv\in E_{2}$.