Related papers: A generalised linear Ramsey graph construction
The purpose of this survey is to provide a gentle introduction to several recent breakthroughs in graph Ramsey theory. In particular, we will outline the proofs (due to various groups of authors) of exponential improvements to the diagonal,…
The set-colouring Ramsey number $R_{r,s}(k)$ is defined to be the minimum $n$ such that if each edge of the complete graph $K_n$ is assigned a set of $s$ colours from $\{1,\ldots,r\}$, then one of the colours contains a monochromatic clique…
The Ramsey's theorem says that a graph with sufficiently many vertices contains a clique or stable set with many vertices. Now we attach some parameter to every vertex, such as degree. Consider the case a graph with sufficiently many…
Given a finite field, one can form a directed graph using the field elements as vertices and connecting two vertices if their difference lies in a fixed subgroup of the multiplicative group. If -1 is contained in this fixed subgroup, then…
An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph…
A sequence of graphs $ \{G_k\} $ is a Ramsey sequence if for every positive integer $ k $, the graph $ G_k $ is a proper subgraph of $ G_{k+1} $, and there exists an integer $n > k$ such that every red-blue coloring of $ G_n $ contains a…
For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is guaranteed…
Given graphs $G, H$ and an integer $q \ge 2$, the generalized Ramsey number, denoted $r(G,H,q)$, is the minimum number of colours needed to edge-colour $G$ such that every copy of $H$ receives at least $q$ colours. In this paper, we prove…
In this paper, we investigate three extensions of Ramsey numbers to other combinatorial settings. We first consider ordered Ramsey numbers. Here, we ask for a monochromatic copy of a linearly ordered graph $G$ in every $2$-edge-coloring of…
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…
Given graphs $H_1,H_2$, a graph $G$ is $(H_1,H_2)$-Ramsey if for every colouring of the edges of $G$ with red and blue, there is a red copy of $H_1$ or a blue copy of $H_2$. In this paper we investigate Ramsey questions in the setting of…
Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if $G$ is an…
Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a…
Analogues of Ramsey's Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic…
Given a $k$-uniform hypergraph $G$ and a set of $k$-uniform hypergraphs $\mathcal{H}$, the generalized Ramsey number $f(G,\mathcal{H},q)$ is the minimum number of colors needed to edge-color $G$ so that every copy of every hypergraph $H\in…
In this note, we introduce a new method for constructing graphs with high chromatic number and small clique. Indeed, via this method, we present a new proof for the well-known Kneser's conjecture.
This paper begins by exploring some old and new results about Ramsey numbers and minimum numbers of monochromatic triangles in $2$-colorings of complete graphs, both in the disjoint and non-disjoint cases. We then extend the theory, by…
Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological…
Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and…
We create a framework for hereditary graph classes $\mathcal{G}^\delta$ built on a two-dimensional grid of vertices and edge sets defined by a triple $\delta=\{\alpha,\beta,\gamma\}$ of objects that define edges between consecutive columns,…