Related papers: Monochromatic Sums and Products over $\mathbb{Q}$
We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd distance from each other.
A decomposition of a natural number n is a sequence of consecutive natural numbers that sums to n. We construct a one-to-one correspondence between the odd factors of a natural number and its decompositions. We study the decompositions by…
A theorem of Galvin asserts that if the unordered pairs of reals are partitioned into finitely many Borel classes then there is a perfect set P such that all pairs from P lie in the same class. The generalization to n-tuples for n >= 3 is…
A coloring of vertices of a graph is called perfect if, for every vertex, the collection of colors of its neighbors depends only on its own color. The correspondent color partition of vertices is called equitable. We note that a number of…
We show that, for every $r, k$, there is an $n = n(r,k)$ so that any $r$-coloring of the edges of the complete graph on $[n]$ will yield a monochromatic complete subgraph on vertices ${a + \sum_{i \in I} d_i \mid I \subseteq [k]}$ for some…
Given a fixed integer $n$, we prove Ramsey-type theorems for the classes of all finite ordered $n$-colorable graphs, finite $n$-colorable graphs, finite ordered $n$-chromatic graphs, and finite $n$-chromatic graphs.
Recently, Hirschhorn and Sellers defined the partition function $a_r(n)$, which counts the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may appear in one of $r$-colors for fixed $r\ge1$. The aim…
A simple graph-product type construction shows that for all natural numbers $r \ge q$, there exists an edge-coloring of the complete graph on $2^r$ vertices using $r$ colors where the graph consisting of the union of arbitrary $q$ color…
We complete a 40-year old program on the computability-theoretic analysis of Ramsey's theorem, starting with Jockusch in 1972, and improving a result of Chong, Slaman and Yang in 2014. Given a set $X$, let $[X]^n$ be the collection of all…
The four-color theorem states that no more than four colors are required to color all nodes in planar graphs such that no two adjacent nodes are of the same color. The theorem was first propounded by Francis Guthrie in 1852. Since then,…
Let $\binom{X}{h}$ be the collection of all $h$-subsets of an $n$-set $X\supseteq Y$. Given a coloring (partition) of a set $S\subseteq \binom{X}{h}$, we are interested in finding conditions under which this coloring is extendible to a…
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…
A constellation pattern is a finite increasing rational sequence \(Q=[0=q_0<q_1<\cdots<q_k=1]\), and a \(Q\)-constellation in \([n]\) is obtained by scaling and translating a rational pattern $Q$, with key examples including arithmetic…
For the four-color theorem that has been developed over one and half centuries, all people believe it right but without complete proof convincing all1-3. Former proofs are to find the basic four-colorable patterns on a planar graph to…
In arXiv:2209.04859 Andy Zucker and Chris Lambie-Hanson proved the consistency result for some coloring principle for the products of polish spaces by at most countable many colors. This principle easy implies Halpern and L\"auchli's…
In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are…
In this short article, we study factor colorings of aperiodic linearly recurrent infinite words. We show that there always exists a coloring which does not admit a monochromatic factorization of the word into factors of increasing lengths.
Independently posed by Behzad and Vizing, the Total Coloring Conjecture asserts that the total chromatic number of a simple connected graph $G$ is either $\Delta(G)+1$ or $\Delta(G)+2$, where $\Delta(G)$ is the largest degree of any vertex…
An infinite graph is highly connected if the complement of any subgraph of smaller size is connected. We consider weaker versions of Ramsey's Theorem asserting that in any coloring of the edges of a complete graph there exist large highly…
A proper $q$-coloring of a domain in $\mathbb{Z}^d$ is a function assigning one of $q$ colors to each vertex of the domain such that adjacent vertices are colored differently. Sampling a proper $q$-coloring uniformly at random, does the…