相关论文: On some problems of Euclidean Ramsey theory
We prove that if one colors each point of the Euclidean plane with one of five colors, then there exist two points of the same color that are either distance $1$ or distance $2$ apart.
It is consistent that ZF+DC holds, the hypergraph of rectangles on a given Euclidean space has countable chromatic number, while the hypergraph of equilateral triangles in two-dimensional Euclidean space does not.
We introduce and study a new type of Ramsey-Turan problems, a typical example of which is the following one: let c>0 and G be a graph of sufficiently large order n with minimum degree >3n/4. If the edges of G are colored in blue or red,…
We consider a generalisation of the classical Ramsey theory setting to a setting where each of the edges of the underlying host graph is coloured with a {\em set} of colours (instead of just one colour). We give bounds for monochromatic…
We study quantitative aspects of the following fact: For every graph $F$, there exists a graph $G$ with the property that any $2$-coloring of the triangles of $G$ yields an induced copy of $F$, in which all triangles are monochromatic. We…
It is consistent relative to an inaccessible cardinal that ZF+DC holds, the hypergraph of equilateral triangles on a given Euclidean space has countable chromatic number, while the hypergraph of isosceles triangles in the plane does not.
In this paper we prove that if $S$ is any finite configuration of points in $\mathbb{Z}^2$, then any finite coloring of $\mathbb{E}^2$ must contain uncountably many monochromatic subsets homothetic to $S$. We extend a result of Brown,…
Conventional Ramsey-theoretic investigations for edge-colourings of complete graphs are framed around avoidance of certain configurations. Motivated by considerations arising in the field of Qualitative Reasoning, we explore edge colourings…
We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd distance from each other.
We study a restriction of Ramsey's theorem for 2-coloring of triples, in which homogeneous sets for color~1 are of bounded size ($\mathsf{BRT}^3_2$). We prove that the computational content of this statement is very close to Ramsey's…
If we two-colour a circle, we can always find an inscribed triangle with angles $(\frac{\pi}{7},\frac{2\pi}{7},\frac{4\pi}{7})$ whose three vertices have the same colour. In fact, Bialostocki and Nielsen showed that it is enough to consider…
It was previously shown that any two-colour colouring of K(C_n) must contain a monochromatic planar K_4 subgraph for n >= N^*, where 6 <= N^* <= N and N is Graham's number. The bound was later improved to 11 <= N^* <= N. In this article, it…
A matching $M$ in a graph $G$ is connected if all the edges of $M$ are in the same component of $G$. Following \L uczak,there have been many results using the existence of large connected matchings in cluster graphs with respect to regular…
Bal and DeBiasio [Partitioning random graphs into monochromatic components, Electron. J. Combin. 24 (2017), Paper 1.18] put forward a conjecture concerning the threshold for the following Ramsey-type property for graphs $G$: every…
A finite Euclidean set is diameter-Ramsey if, for every number of colors, some finite same-diameter witness has the property that every coloring of the witness contains a monochromatic congruent copy of the set. Frankl, Pach, Reiher and…
We present a short ultrafilter proof of the existence of monochromatic exponential triples $\{a, b, b^a\}$ in any finite coloring of the natural numbers. The proof is given from scratch and uses only Ramsey's theorem, the notion of…
Consider the graph $\mathbb{H}(d)$ whose vertex set is the hyperbolic plane, where two points are connected with an edge when their distance is equal to some $d>0$. Asking for the chromatic number of this graph is the hyperbolic analogue to…
A measure theoretic approach of the problem that there exits a finite unit-distance graphs in the plane that are not five (or four) colorable.
In this short note we prove that there is a constant $c$ such that every k-edge-coloring of the complete graph K_n with n > 2^{ck} contains a K_4 whose edges receive at most two colors. This improves on a result of Kostochka and Mubayi, and…
Ramsey's theorem, concerning the guarantee of certain monochromatic patterns in large enough edge-coloured complete graphs, is a fundamental result in combinatorial mathematics. In this work, we highlight the connection between this…