Related papers: Coloring trees in reverse mathematics
Connection of the Four Color Theorem (FCT) with some operations on trees is described. L.H. Kauffman's theorem about FCT and vector cross product is discussed. Operation of transplantation on trees linked with the move of brackets according…
We prove that double exponentiation is an upper bound to Ramsey theorem for colouring of pairs when we want to predetermine the order of the differences of successive members of the homogeneous set.
Ramsey's theorem states that for any coloring of the n-element subsets of N with finitely many colors, there is an infinite set H such that all n-element subsets of H have the same color. The strength of consequences of Ramsey's theorem has…
In \cite{J}, Theorem 4.2, Jockusch proves that for any computable k-coloring of pairs of integers, there is an infinite $\Pi^0_2$ homogeneous set. The proof uses a countable collection of $\Pi^0_2$ sets as potential infinite homogeneous…
We consider the restriction of Ramsey's theorem that arises from considering only translation-invariant colourings of pairs, and show that this has the same strength (both from the viewpoint of Reverse Mathematics and from the viewpoint of…
The first-order part of the Ramsey's Theorem for pairs with an arbitrary number of colors is known to be precisely BSigma03. We compare this to the known division of Ramsey's Theorem for pairs into the weaker principles, EM (the…
We show that for every finite colouring of the natural numbers there exists $a,b >1$ such that the triple $\{a,b,a^b\}$ is monochromatic. We go on to show the partition regularity of a much richer class of patterns involving exponentiation.…
A Ramsey-like theorem is a statement of the form ``For every 2-coloring of $[\mathbb{N}]^2$, there exists an infinite set~$H \subseteq \mathbb{N}$ such that $[H]^2$ avoids some pattern''. We prove that none of these statements are…
This paper falls within the general program of investigating the proof theoretic strength (in terms of reverse mathematics) of combinatorial principals which follow from versions of Ramsey's theorem. We examine two statements in graph…
We show that there exists a constant $c>0$ such that every $n$-vertex tree $T$ with $\Delta(T)\le cn$ has Ramsey number $R(T)=\max\{t_1+2t_2,2t_1\}-1$, where $t_1\ge t_2$ are the sizes of the bipartition classes of $T$. This improves an…
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…
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…
We formulate and prove the generalizations of Friedman's free set and thin set theorems and of the rainbow Ramsey theorem to colorings of barriers. We analyze the strength of these theorems from the point of view of computability theory…
Every function over the natural numbers has an infinite subdomain on which the function is non-decreasing. Motivated by a question of Dzhafarov and Schweber, we study the reverse mathematics of variants of this statement. It turns out that…
We prove a general Ramsey theorem for trees with a successor operation. This theorem is a common generalization of the Carlson-Simpson Theorem and the Milliken Tree Theorem for regularly branching trees. Our theorem has a number of…
Let $\mathfrak M=(M,\mathcal X)$ be a model of $\mathsf{RCA}_0+\text{$\Sigma^0_2$-bounding}$ in which $\Sigma^0_2(A)$-induction fails for some $A\in\mathcal X$. We show that (i) if $\mathfrak M$ is a model of the combinatorial principle…
The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic. In this setting, called reverse mathematics, one investigates which theorems provably imply which others in a…
Ramsey's theorem asserts that every $k$-coloring of $[\omega]^n$ admits an infinite monochromatic set. Whenever $n \geq 3$, there exists a computable $k$-coloring of $[\omega]^n$ whose solutions compute the halting set. On the other hand,…
Hirst investigated a slight variant of Hindman's Finite Sums Theorem -- called Hilbert's Theorem -- and proved it equivalent over $\RCA_0$ to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural…
We study the first-order consequences of Ramsey's Theorem for $k$-colourings of $n$-tuples, for fixed $n, k \ge 2$, over the relatively weak second-order arithmetic theory $\mathrm{RCA}^*_0$. Using the Chong-Mourad coding lemma, we show…