On Ramsey-Minimal Infinite Graphs

Procedia Computer Science, 2015

For any graphs G and H, we write F → (G, H) to means that in any red-blue coloring of all the edges of F, the graph F will contain either a red G or a blue H. A graph F is called a Ramsey (G,H)-minimal graph if F satisfies two conditions: F → (G, H), and F * (G, H) for every subgraph F * of F. The set of all Ramsey (G, H)-minimal graphs is denoted by R(G, H). In this paper, we construct some family of graphs which belong to R(P 3 , P n ), for any n ≥ 6. In particular, we give an infinite class of trees which provides Ramsey (P 3 , P 7 )-minimal graphs.

2010

Given a fixed integer $n$, we prove Ramsey-type theorems for the classes of all finite ordered $n$-colorable graphs, finite $n$-colorable graphs, finite ordered $n$-chromatic graphs, and finite $n$-chromatic graphs.

Discrete Mathematics, 1982

Journal of Combinatorial Theory, Series B, 2016

A graph G is r-Ramsey for a graph H, denoted by G → (H) r , if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramseyminimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property. Let s r (H) denote the smallest minimum degree of G over all graphs G that are r-Ramseyminimal for H. The study of the parameter s 2 was initiated by Burr, Erdős, and Lovász in 1976 when they showed that for the clique s 2 (K k) = (k − 1) 2. In this paper, we study the dependency of s r (K k) on r and show that, under the condition that k is constant, s r (K k) = r 2 • polylog r. We also give an upper bound on s r (K k) which is polynomial in both r and k, and we determine s r (K 3) up to a factor of log r.

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.

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.

Applied Mathematical Sciences, 2015

Let G and H be two given graphs. The notation F → (G, H) means that any red-blue coloring on the edges of F will create either a red subgraph G or a blue subgraph H in F. A graph F is a Ramsey (G, H)minimal graph if F → (G, H) and F * → (G, H) for any proper subgraph F * ⊂ F. The set of all (G, H)-minimal graphs is denoted by R(G, H). In this paper we give some necessary conditions for the members of R(2K 2 , 2C n) for n ≥ 3 and determine some graphs in R(2K 2 , 2C 3) and R(2K 2 , 2C 4).

Duke Mathematical Journal, 2013

Ramsey's theorem, in the version of Erdős and Szekeres, states that every 2-coloring of the edges of the complete graph on {1, 2,. .. , n} contains a monochromatic clique of order 1 2 log n. In this paper, we consider two well-studied extensions of Ramsey's theorem. Improving a result of Rödl, we show that there is a constant c > 0 such that every 2-coloring of the edges of the complete graph on {2, 3, ..., n} contains a monochromatic clique S for which the sum of 1/ log i over all vertices i ∈ S is at least c log log log n. This is tight up to the constant factor c and answers a question of Erdős from 1981. Motivated by a problem in model theory, Väänänen asked whether for every k there is an n such that the following holds. For every permutation π of 1,. .. , k − 1, every 2-coloring of the edges of the complete graph on {1, 2,. .. , n} contains a monochromatic clique a 1 <. .. < a k with a π(1)+1 − a π(1) > a π(2)+1 − a π(2) >. .. > a π(k−1)+1 − a π(k−1). That is, not only do we want a monochromatic clique, but the differences between consecutive vertices must satisfy a prescribed order. Alon and, independently, Erdős, Hajnal and Pach answered this question affirmatively. Alon further conjectured that the true growth rate should be exponential in k. We make progress towards this conjecture, obtaining an upper bound on n which is exponential in a power of k. This improves a result of Shelah, who showed that n is at most double-exponential in k.

Discrete Mathematics, 2001

For a graph G let RM(G) be the smallest integer R, if it exists, such that every coloring of the edges of KR by an arbitrary number of colors implies a subgraph of KR isomorphic to G that is either monochromatic or has the property that no two of its edges have the same color. We generalize the theorem r(nK2; n − 1) = n 2 − n + 2, where r(nK2; n − 1) is the Ramsey number for n matchings in an (n − 1)-coloring of the complete graph. Namely, we prove that RM(nK2) = r(nK2; n − 1) = n 2 − n + 2. In addition, we generalize the theorem r(nK2; 2) = 3n − 1 by considering colorings with three and ÿve colors. Several further possible generalizations for hypermatchings in hypergraphs are suggested.

arXiv (Cornell University), 2015

We extend two well-known results in Ramsey theory from from K n to arbitrary n-chromatic graphs. The first is a note of Erdős and Rado stating that in every 2-coloring of the edges of K n there is a monochromatic tree on n vertices. The second is the theorem of Cockayne and Lorimer stating that for positive integers satisfying n 1 = max{n 1 , n 2 , . . . , n t } and with n = n 1 +1+ t i=1 (n i -1), the following holds. In every coloring of the edges of K n with colors 1, 2 . . . , t there is a monochromatic matching of size n i for some i ∈ {1, 2, . . . , t}.

COMBINATORICA, 1998

We investigate the induced Ramsey number r ind (G, H) of pairs of graphs (G, H). This number is defined to be the smallest possible order of a graph Γ with the property that, whenever its edges are coloured red and blue, either a red induced copy of G arises or else a blue induced copy of H arises. We show that, for any G and H with k = |V (G)| ≤ t = |V (H)|, we have

Proceedings of the American Mathematical Society, 1992

We present a new proof of the Paris-Harrington unprovable (in PA) version of Ramsey's theorem. This also yields a particularly short proof of the Ketonen-Solovay result on rapidly growing Ramsey functions.

Discrete Mathematics, 2001

For each vertex s of the subset S of vertices of a graph G, we deÿne Boolean variables p; q; r which measure the existence of three kinds of S-private neighbours (S-pns) of s. A 3-variable Boolean function f = f(p; q; r) may be considered as a compound existence property of S-pns. The set S is called an f-set of G if f = 1 for all s ∈ S and the class of all f-sets of G is denoted by f . Special cases of f include the independent sets, irredundant sets and CO-irredundant sets of G. For some f ∈ F it is possible to deÿne analogues (involving f-sets) of the classical Ramsey graph numbers. We consider existence theorems for these f-Ramsey numbers and prove that some of them satisfy the well-known recurrence inequality which holds for the classical Ramsey numbers.

The Mathematical Intelligencer, 2007

BASEL, BIRKHAUSER, 2005, PP 257, C38, ISBN 3-7643-7264-8 REVIEWED BY HANS-PETER A. KLINZI Ihe book introduces the reader i ] to sophisticated Ramsey-theoretic I methods that have recently been used in the theo W of Banach spaces.

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...