REPRESENTATIONS OF RAMSEY RELATION ALGEBRAS

Notre Dame Journal of Formal Logic

Hindman's Theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. A Ramsey algebra is a structure that satisfies an analogue of Hindman's Theorem. In this paper, we present the basic notions of Ramsey algebras using terminology from mathematical logic. We also present some results regarding classification of Ramsey algebras.

2018

The study of Ramsey algebras is a Ramseyan-type study on algebras. The precise formulation of a Ramsey algebra is based on the work of Carlson on topological Ramsey spaces, from which a wide array of classical combinatorial results such as the Ellentuck theorem and Hindman’s theorem can be derived. After his groundbreaking work on Ramsey spaces, Carlson suggested that, for spaces that are generated by algebras, one may pursue a purely combinatorial study of these spaces, where results of topological nature can be derived from their associated combinatorial results. Such a direction of study would then be known as Ramsey algebra. The suggestion was first pursued by Teh in his doctoral work and some basic results concerning homogeneous algebras were obtained

In this paper, we are interested in the problem of evaluation of the classical multicolor Ramsey number R(3,3,3). We first convert it successfully into asystem of clauses of 3-literals each, i.e., a 3-SAT instance. We then describe the algebraic method of Greenwood and Gleason [2], which is based on the finite field F 16 , that constructs a monochromatic triangle-free edge-coloring with three colors of the Ramsey graph K 16 associated with the number R(3,3,3). We propose a simple and new coloring method, completely different from that of Greenwood and Gleason, which colors the edges of the Ramsey graph K 16 with three colors without any monochromatic triangle.

Graphs and Combinatorics

We prove induced Ramsey theorems in which the monochromatic induced subgraph satisfies that all members of a prescribed set of its partial isomorphisms extend to automorphisms of the colored graph (without requirement of preservation of colors).

The notion of a topological Ramsey space was introduced by Carlson some 30 years ago. Studying the topological Ramsey space of variable words, Carlson was able to derive many classical combinatorial results in a unifying manner. For the class of spaces generated by algebras, Carlson had suggested that one should attempt a purely combinatorial approach to the study. This approach was later formulated and named Ramsey algebra. In this paper, we continue to look at heterogeneous Ramsey algebras, mainly characterizing various Ramsey algebras involving matrices.

Carlson introduced the notion of a Ramsey space as a generalization to the Ellentuck space. When a Ramsey space is induced by an algebra, Carlson suggested a study of its purely com-binatorial version now called Ramsey algebra. Some basic results for homogeneous algebras have been obtained. In this paper, we introduce the notion of a Ramsey algebra for heterogeneous algebras and derive some basic results. Then, we study the Ramsey-algebraic properties of vector spaces.

European Journal of Combinatorics, 1985

A class J{ of relational systems (of the same type) has the 8-Ramsey property if for every $ E J{ there is T E J{ such that to every 2-coloring of (!) (= relational subsystems of T isomorphic to 8) we can find a monochromatic (iJ for some {J E (D. Extending recent results by JeZek and Ndetnl we prove it for (a) every class J{ of finite reflexive relational systems closed for products and 8 E J{ a singleton, (b) every abstract class J{ of finite relational systems with the strong amalgamation property and 8 E J{ such that the sets from (!) are disjoint for all $ E J{. Finally we prove: Let J{ be an abstract class of finite reflexive or areflexive relational systems with the strong amalgamation property. If J{ has the 8-Ramsey property, then 8 is constant.

Discrete Mathematics, 2016

A relational structure R is rainbow Ramsey if for every finite induced substructure C of R and every colouring of the copies of C with countably many colours, such that each colour is used at most k times for a fixed k, there exists a copy R * of R so that the copies of C in R * use each colour at most once. We show that certain ultrahomogenous binary relational structures, for example the Rado graph, are rainbow Ramsey. Via compactness this then implies that for all finite graphs B and C and k ∈ ω, there exists a graph A so that for every colouring of the copies of C in A such that each colour is used at most k times, there exists a copy B * of B in A so that the copies of C in B * use each colour at most once.

Discrete Mathematics, 1982

Journal of Mathematical Logic, 2019

The universal homogeneous triangle-free graph, constructed by Henson [A family of countable homogeneous graphs, Pacific J. Math. 38(1) (1971) 69–83] and denoted [Formula: see text], is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with Erdős–Hajnal–Posá [Strong embeddings of graphs into coloured graphs, in Infinite and Finite Sets. Vol.[Formula: see text] , eds. A. Hajnal, R. Rado and V. Sós, Colloquia Mathematica Societatis János Bolyai, Vol. 10 (North-Holland, 1973), pp. 585–595] and culminating in work of Sauer [Coloring subgraphs of the Rado graph, Combinatorica 26(2) (2006) 231–253] and Laflamme–Sauer–Vuksanovic [Canonical partitions of universal structures, Combinatorica 26(2) (2006) 183–205], the Ramsey theory of [Formula: see text] had only progressed to bounds for vertex colorings [P. Komjáth and V. Rödl, Coloring of universal graphs, Graphs Combin. 2(1) (1986) 55–60] and edge colorings [N....