A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number, and Ramsey saturated otherwise. We present some conjectures and results concerning both Ramsey saturated and unsaturated graphs. In particular, we show that cycles Cn and paths Pn on n vertices are Ramsey unsaturated for all n ≥ 5. 1 Results and Conjectures Throughout this article, r...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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...

We show that, assuming GCH, if κ is a Ramsey or a strongly Ramsey cardinal and F is a class function on the regular cardinals having a closure point at κ and obeying the constraints of Easton’s theorem, namely, F (α) ≤ F (β) for α ≤ β and α < cf(F (α)), then there is a cofinality preserving forcing extension in which κ remains Ramsey or strongly Ramsey respectively and 2δ = F (δ) for every regu...

Let G and H be simple graphs. The Ramsey number r(G,H) for a pair of graphs G and H is the smallest number r such that any red-blue coloring of the edges of Kr contains a red subgraph G or a blue subgraph H . The size Ramsey number r̂(G,H) for a pair of graphs G and H is the smallest number r̂ such that there exists a graph F with size r̂ satisfying the property that any red-blue coloring of the e...

In this article we give the generalized triangle Ramsey numbers R(K3, G) of 12 005 158 of the 12 005 168 graphs of order 10. There are 10 graphs remaining for which we could not determine the Ramsey number. Most likely these graphs need approaches focusing on each individual graph in order to determine their triangle Ramsey number. The results were obtained by combining new computational and th...

Following ideas of Richer (2000) we introduce the notion of unordered regressive Ramsey numbers or unordered Kanamori-McAloon numbers. We show that these are of Ackermannian growth rate. For a given number-theoretic function f we consider unordered f -regressive Ramsey numbers and classify exactly the threshold for f which gives rise to the Ackermannian growth rate of the induced unordered f -r...

We present a general method for constructing a new class of topological Ramsey spaces. Members of such spaces are infinite sequences of products of Fräıssé classes of finite relational structures satisfying the Ramsey property. We extend the Product Ramsey Theorem of Sokič to equivalence relations for finite products of structures from Fräıssé classes of finite relational structures satisfying ...

A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number, and Ramsey saturated otherwise. This has been studied by Balister, Lehel and Schelp [J. Graph Theory 51 (2006), 22–32]. In this paper, we show that some circulant graphs, trees with diameter 3, and Kt,n ∪ mK1 for infinitely many t, n and m, are Ramsey unsaturated.