The paper gives an account of previous and recent attempts to determine the order of a smallest graph not containing K5 and such that every 2-coloring of its edges results in a monochromatic triangle. A new 14-vertex K4-free graph with the same Ramsey property in the vertex coloring case is found. This yields a new construction of one of the only two known 15-vertex (3,3)-Ramsey graphs not cont...

The problem of explicitly constructing Ramsey graphs, i.e graphs that do not have a large clique or independent set is considered. We provide an elementary construction of a graph with the property that there is no clique or independent set of t of nodes, while the graph size is t p log log t log log log t . The construction is based on taking the product of all graphs in a distribution that is...

The planar Ramsey number PR(G, H) is defined as the smallest integer n for which any 2-colouring of edges of Kn with red and blue, where red edges induce a planar graph, leads to either a red copy of G, or a blue H. In this note we study the weak induced version of the planar Ramsey number in the case when the second graph is complete.

c n (F ) = | V (F ) | and e (F ) = | E (F ) | . The graph F  denotes the complement of F . A graph F will be alled a (G , H )−good graph, if F does not contain G and F  does not contain H . Any (G , H )-good s t graph on n vertices will be called a (G , H , n )−good graph. The Ramsey number R (G , H ) is defined a he smallest integer n such that no (G , H , n )-good graph exists. Any graph is...

For given finite (unordered) graphs G and H, we examine the existence of a Ramsey graph F for which the strong Ramsey arrow F −→ (G)r holds. We concentrate on the situation when H is not a complete graph. The set of graphs G for which there exists an F satisfying F −→ (G)2 2 (P2 is a path on 3 vertices) is found to be the union of the set of chordal comparability graphs together with the set of...

An (r, b)-graph is a graph that contains no clique of size r and no independent set of size b. The set of extremal Ramsey graphs ERG(r, b) consists of all (r, b)-graphs with R(r, b) − 1 vertices, where R(r, b) is the classical Ramsey number. We show that any G ∈ ERG(r, b) is r − 1 vertex connected and 2r − 4 edge connected for r, b ≥ 3.

A set X of vertices of a graph G is said to be 1-dependent if the subgraph of G induced by X has maximum degree one. The 1-dependent Ramsey number t1(l, m) is the smallest integer n such that for any 2-edge colouring (R, B) of Kn, the spanning subgraph B of Kn has a 1-dependent set of size l or the subgraph R has a 1-dependent set of size m. The 2-edge colouring (R, B) is a t1(l, m) Ramsey colo...

Let KN denote the complete k-graph on N vertices, that is, the k-uniform hypergraph whose edges are the (k) k-subsets of a set of N vertices . The Ramsey number of a pair of k-graphs is defined as for graphs in terms of the 2-colorings of the k-sets which are the edges of a complete k-graph . Let sKk denote a k-graph which is a matching with s edges . That is, sKk has sk vertices and s edges, n...

Let G1 and G2 be two given graphs. The Ramsey number R(G1,G2) is the least integer r such that for every graph G on r vertices, either G contains a G1 or G contains a G2. A complete bipartite graph K1,n is called a star. The kipas ̂ Kn is the graph obtained from a path of order n by adding a new vertex and joining it to all the vertices of the path. Alternatively, a kipas is a wheel with one edg...

A graph on n vertices is said to be C-Ramsey if every clique or independent set of the graph has size at most C log n. The only known constructions of Ramsey graphs are probabilistic in nature, and it is generally believed that such graphs possess many of the same properties as dense random graphs. Here, we demonstrate one such property: for any fixed C > 0, every C-Ramsey graph on n vertices i...