Generalized Ramsey theory and decomposable properties of graphs

We consider the numbers associated with Ramsey's theorem as it pertains to partitions of the pairs of elements of a set into two classes. Our purpose is to give a unified development of enumerative techniques which give sharp upper bounds on these numbers and to give constructive methods for partitions to determine lower bounds on these numbers. Explicit computations include the values of R(3, 6) and R(3, 7) among others. Our computational techniques yield the upper bound R(x, y) < cyX-llog log y/log y for x ~> 3.

Discrete Mathematics, 2007

The product P•Q of graph properties P, Q is a class of all graphs having a vertex-partition into two parts inducing subgraphs with properties P and Q, respectively. For a graph invariant ϕ and a graph property P we define ϕ(P) as the minimum of ϕ(F ) taken over all minimal forbidden subgraphs F of P. An invariant of graph properties ϕ is said to be additive with respect to reducible hereditary properties if there is a constant c such that ϕ(P •Q) = ϕ(P)+ϕ(Q)+c for every pair of hereditary properties P, Q. In this paper we provide a necessary and sufficient condition for invariants that are additive with respect to reducible hereditary graph properties. We prove that the order of the largest tree, the chromatic number, the colouring number, the tree-width and some other invariants of hereditary graph properties are of such type.

Pacific Journal of Mathematics, 1977

A graph G is said to have a factorization into the subgraphs G u-, G k if the subgraphs are spanning, pairwise edge-disjoint, and the union of their edge sets equals the edge set of G. For a graphical parameter / and positive integers rti, n 2 , , n k (k ^ 1), the /-Ramsey number f{n u n 2 , , n k) is the least positive integer p such that for any factorization K p = Uΐ=iG l9 it follows that /(G)^n t for at least one i, l^i^k. In the following, we present two results involving /-Ramsey numbers which hold for various vertex and edge partition parameters, respectively. It is then shown that the concept of /-Ramsey number can be generalized to more than one vertex partition parameter, more than one edge partition parameter, and combinations of vertex and edge partition parameters. Formulas are presented for these generalized /-Ramsey numbers and specific illustrations are given.

Discussiones Mathematicae Graph Theory, 1995

Let IL be the set of all hereditary and additive properties of graphs. For P 1 , P 2 ∈ IL, the reducible property R = P 1 • P 2 is defined as follows: G ∈ R if and only if there is a partition V (G) = V 1 ∪ V 2 of the vertex set of G such that V 1 G ∈ P 1 and V 2 G ∈ P 2. The aim of this paper is to investigate the structure of the reducible properties of graphs with emphasis on the uniqueness of the decomposition of a reducible property into irreducible ones.

Discrete Mathematics, 2004

Discrete Mathematics, 2004

Let P be a property of graphs. A graph G is vertex (P; k)-colourable if the vertex set V (G) of G can be partitioned into k sets V1; V2; : : : ; V k such that the subgraph G[Vi] of G belongs to P, i = 1; 2; : : : ; k. If P is a hereditary property, then the set of minimal forbidden subgraphs of P is deÿned as follows: F(P) = {G: G ∈ P but each proper subgraph H of G belongs to P}. In this paper we investigate the property On: each component of G has at most n+1 vertices. We construct minimal forbidden subgraphs for the property (O k n) "to be (On; k)-colourable". We write G v → (H) k , k ¿ 2, if for each k-colouring V1; V2; : : : ; V k of a graph G there exists i, 1 6 i 6 k, such that the graph induced by the set Vi contains H as a subgraph. A graph G is called (H) k-vertex Ramsey minimal if G v → (H) k , but G v 9 (H) k for any proper subgraph G of G. The class of (P3) k-vertex Ramsey minimal graphs is investigated.

Bulletin of the London Mathematical Society, 1974

Continuing this series of papers on generalized Ramsey theory for graphs, we define the Ramsey number r(D lt D 2) of two digraphs D t and D 2 as the minimum p such that every 2-colouring of the arcs (directed lines) of DK P (the complete symmetric digraph of order p) contains a monochromatic D t or D 2. It is shown (Theorem 1) that this number exists if and only if D y or D 2 is acyclic. Then r(D), the diagonal Ramsey number of a given acyclic digraph D, is defined as r(D, D). Notation: D' is the converse of D, GD is the underlying graph of D, DG is the symmetric digraph of G, and T p is the transitive tournament of order p. Let r(m, n) be the traditional Ramsey number of the two complete graphs K m and K n. Finally, let S n be the star with n arcs from one point u to n points v r Assuming the Ramsey numbers under discussion exist, we prove the following results: THEOREM 2. r(D u D 2) = r(D x ', D 2). THEOREM 3. r(D u D 2) ^ r(GD x , GD 2). THEOREM 4. r(D u D 2) < r(T Pl , T P2) if both D t and D 2 (with p x and p 2 points respectively) are acyclic.

1996

Agsrnacr. The set Ramsey number r.(t, (l)) is the smallest integer r such that if the edges of a complete graph K, are 2-colored, then there will be a graph with n vertices a,nd /c edges in the first cblorbr a graph with n vertices ana (|) (e.g. a complete graph) in the second eolor. For each n ) 3 and 1 < k < n, the set Ramsey numbers r.(f, (l)) are determined. One approach to get some insight into r(K") was suggested in , where Ramsey numbers for sets of graphs with fixed numbers of vertices and edges were considered. Thus, the following definition. Definition 1. For positive integers n'L,r\s,t with 1 ( s ( (\) ana t < t < (Z) the set Ramsey number r^,n(s,t) is the smallest integer r such that for arry 2-coloring of the edges of a complete graph K,, there is either a graph with m vertices and's edges in the first color, or a graph with n vertices and t edges in the second color. When rrl : rL, r*,^(s,t) will be expressed more compactly as r*(s,t). Associated with fixed positive integers rn and n there is an (!) UV (i) array of Ramsey numbers (r*,n(t,t)) for 1 ( s ( (f) ana 1 <, < (!) ttrat represent sets of graphs with fixed numbers of edges. For small values of m and n the array of Ramsey numbers (r*,n(t,t)) have been determined. The values for m :3 and 3 I n 17 were determined in [4], and the values for m: 4 and 4 1n 15 were determined in , except ior ra,s((l), (!)) : r(Ka, K5), which has now been shown to be 25 by McKay and R^adziszowski (see [fl). All but 5 cf the values of (r5(s,t)) were determined in [5]. For Ramsey numbers of more general sets of small graphs see [2].

European Journal of Combinatorics, 2010

2021

A set of vertices X ⊆ V in a simple graph G(V, E) is irredundant if each vertex x ∈ X is either isolated in the induced subgraph X or else has a private neighbor y ∈ V \ X that is adjacent to x and to no other vertex of X. The irredundant Ramsey number s(m, n) is the smallest N such that in every red-blue coloring of the edges of the complete graph of order N , either the blue subgraph contains an m-element irredundant set or the red subgraph contains an n-element irredundant set. The mixed Ramsey number t(m, n) is the smallest N for which every red-blue coloring of the edges of K N yields an m-element irredundant set in the blue subgraph or an n-element independent set in the red subgraph. In this paper, we first improve the upper bound of t(3, n); using this result, we confirm that a conjecture proposed by Chen, Hattingh, and Rousseau, that is, lim n→∞ t(m,n) r(m,n) = 0 for each fixed m ≥ 3, is true for m ≤ 4. At last, we prove that s(3, 9) and t(3, 9) are both equal to 26.

Let H be a graph with the chromatic number χ(H) and the chromatic surplus σ(H). A connected graph G of order n is called good with respect to H, denoted by H-good, if R(G, H) = (n−1)(χ(H)−1)+σ(H). In this paper, we investigate the Ramsey numbers for a union of graphs not necessarily containing an H-good component.

Cambridge University Press eBooks, 2018

We state the Ramsey property of classes of ordered structures with closures and given local properties. This generalises many old and new results: the Nešetřil-Rödl Theorem, the authors Ramsey lift of bowtie-free graphs as well as the Ramsey Theorem for Finite Models (i.e. structures with both functions and relations) thus providing the ultimate generalisation of Structural Ramsey Theorem. We give here a more concise reformulation of the recent paper "All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms)" and the main purpose of this paper is to show several applications. Particularly we prove the Ramsey property of ordered sets with equivalences on the power set, Ramsey theorem for Steiner systems, Ramsey theorem for resolvable designs and a partial Ramsey type results for H-factorizable graphs. All of these results are natural, easy to state, yet proofs involve most of the theory developed.

arXiv: Combinatorics, 2019

Given a graph $G$, a $k$-sparse $j$-set is a set of $j$ vertices inducing a subgraph with maximum degree at most $k$. A $k$-dense $i$-set is a set of $i$ vertices that is $k$-sparse in the complement of $G$. As a generalization of Ramsey numbers, the $k$-defective Ramsey number $R_k^{\mathcal{G}}(i,j)$ for the graph class $\mathcal{G}$ is defined as the smallest natural number $n$ such that all graphs on $n$ vertices in the class $\mathcal{G}$ have either a $k$-dense $i$-set or a $k$-sparse $j$-set. In this paper, we examine $R_k^{\mathcal{G}}(i,j)$ where $\mathcal{G}$ represents various graph classes. In forests and cographs, we give formulas for all defective Ramsey numbers. In cacti, bipartite graphs and split graphs, we provide defective Ramsey numbers in most of the cases and point out open questions, formulated as conjectures if possible.

Journal of Mathematical Logic

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 substructure rather than one color, as that is often impossible. Such theorems for Henson graphs however remained elusive, due to lack of techniques for handling forbidden cliques. Building on the author’s recent result for the triangle-free Henson graph, we prove that for each [Formula: see text], the k-clique-free Henson graph has finite big Ramsey degrees, the appropriate analogue of Ramsey’s Theorem. We develop a method for coding copies of Henson graphs into a new class of trees, called strong coding trees, and prove Ramsey theorems for these trees which are applied to deduce finite big Ramsey degrees. The approach here provides a general methodology opening further study of big Ramsey degrees for ultrahomogeneous structures. The results have bearing o...

2005

We explore the properties of subgraphs (called Markovian subgraphs) of a decomposable graph under some condition. For a decomposable graph G and a collection γ of its Markovian subgraphs, we show that the set χ(G) of the intersections of all the neighboring cliques of G contains ∪g∈γχ(g). We also show that χ(G) = ∪g∈γχ(g) holds for a certain type of G which we call a maximal Markovian supergraph of γ.