The irredundant Ramsey number s = s(m,n) [upper domination Ramsey number u = u(m,n), respectively] is the smallest natural number s [u, respectively] such that in any red-blue edge colouring (R,B) of the complete graph of order s [u, respectively], it holds that IR(B) ≥ m or IR(R) ≥ n [Γ(B) ≥ m or Γ(R) ≥ n, respectively], where Γ and IR denote respectively the upper domination number and the ir...

We study two classical problems in graph Ramsey theory, that of determining the Ramsey number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph with a given number of vertices. The Ramsey number r(H) of a graph H is the least positive integer N such that every twocoloring of the edges of the complete graph KN contains a monochromatic copy of H. A famous resul...

For a k-uniform hypergraph G with vertex set {1, . . . , n}, the ordered Ramsey number ORt(G) is the least integer N such that every t-coloring of the edges of the complete k-uniform graph on vertex set {1, . . . , N} contains a monochromatic copy of G whose vertices follow the prescribed order. Due to this added order restriction, the ordered Ramsey numbers can be much larger than the usual gr...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii PRELIMINARIES AND NOTATION . . . . . . . . . . . . . . . . . . . . . ix CHAPTER 1. OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The Graph Ramsey Number . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Arrow Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 The Dir...

The upper domination Ramsey number u(m,n) is the smallest integer p such that every 2-coloring of the edges of Kp with color red and blue, Γ(B) ≥ m or Γ(R) ≥ n, where B and R is the subgraph of Kp induced by blue and red edges, respectively; Γ(G) is the maximum cardinality of a minimal dominating set of a graph G. In this paper, we show that u(4, 4) ≤ 15.

We say that a graph G has the Ramsey property w.r.t. some graph F and some integer r ≥ 2, or G is (F, r)-Ramsey for short, if any r-coloring of the edges of G contains a monochromatic copy of F . Rödl and Ruciński asked how globally sparse (F, r)-Ramsey graphs G can possibly be, where the density of G is measured by the subgraph H ⊆ G with the highest average degree. So far, this so-called Rams...

In 2005, Kechris, Pestov and Todorčević provided a powerful tool to compute an invariant of topological groups known as the universal minimal flow, immediately leading to an explicit representation of this invariant in many concrete cases. More recently, the framework was generalized allowing for further applications, and the purpose of this paper is to apply these new methods in the context of...

The nonclassical mixed domination Ramsey number v(m,G) is the smallest integer p such that in every 2-coloring of the edges of Kp with color red and blue, either Γ(B) ≥ m or there exists a blue copy of graph G, where B is the subgraph of Kp induced by blue edges. Γ(G) is the maximum cardinality of a minimal dominating set of a graph G. We give exact values for numbers v(m,K3 − e), v(3, Pm), v(3...