The Ramsey multiplicity R(G) of a graph G is the minimum number of monochromatic copies of G in any two-colouring of the edges of Kr(G), where r(G) denotes the Ramsey number of G. Here we prove that odd cycles have super-exponentially large Ramsey multiplicity: If Cn is an odd cycle of length n, then logR(Cn) = Θ(n logn).

Abstract The size-Ramsey number of a graph F is the smallest edges in G with Ramsey property for , that is, any 2-colouring contains monochromatic copy . We prove grid on n × vertices bounded from above by 3+o(1)

The Ramsey number r(G) of a graph G is the minimum N such that every red–blue coloring of the edges of the complete graph on N vertices contains a monochromatic copy of G. Determining or estimating these numbers is one of the central problems in combinatorics. One of the oldest results in Ramsey Theory, proved by Erdős and Szekeres in 1935, asserts that the Ramsey number of the complete graph w...

For given graphs G and H, the Ramsey number R(G,H) is the least natural number n such that for every graph F of order n the following condition holds: either F contains G or the complement of F contains H. In this paper, we determine the Ramsey number of paths versus generalized Jahangir graphs. We also derive the Ramsey number R(tPn,H), where H is a generalized Jahangir graph Js,m where s ≥ 2 ...

It was conjectured by Paul Erdős that if G is a graph with chromatic number at least k, then the diagonal Ramsey number r(G) ≥ r(Kk). That is, the complete graph Kk has the smallest diagonal Ramsey number among the graphs of chromatic number k. This conjecture is shown to be false for k = 4 by verifying that r(W6) = 17, where W6 is the wheel with 6 vertices, since it is well known that r(K4) = ...

We say that a graph F strongly arrows (G, H) and write F ֌ (G, H) if for every edge-coloring of F with colors red and blue a red G or a blue H occurs as an induced subgraph of F . Induced Ramsey numbers are defined by r(G, H) = min{|V (G)| : F ֌ (G, H)}. The value of r(G, H) is finite for all graphs, and good upper bounds on induced Ramsey numbers in general, and for particular families of grap...

We provide an elementary proof of the fact that the ramsey number of every bipartite graph H with maximum degree at most ∆ is less than 8(8∆)|V (H)|. This improves an old upper bound on the ramsey number of the n-cube due to Beck, and brings us closer toward the bound conjectured by Burr and Erdős. Applying the probabilistic method we also show that for all ∆≥1 and n≥∆+1 there exists a bipartit...

For given graphs G and H, the Ramsey number R(G, H) is the least natural number n such that for every graph F of order n the following condition holds: either F contains G or the complement of F contains H. In this paper, we determine the Ramsey number of paths versus generalized Jahangir graphs. We also derive the Ramsey number R(tPn, H), where H is a generalized Jahangir graph Js,m where s ≥ ...

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The weight of a graph is defined as the sum of the weights of its edges. In 2-edge-colored complete graph, by using Ramsey-type theorems, we obtain the existence of monochromatic subgraph which have many edges compared with its order. In this paper, we extend the concept of Ramsey problem to ...

A new upper bound is given for the cycle-complete graph Ramsey number r(C,,,, K„), the smallest order for a graph which forces it to contain either a cycle of order m or a set of ri independent vertices . Then, another cycle-complete graph Ramsey number is studied, namely r( :C,,,, K„) the smallest order for a graph which forces it to contain either a cycle of order 1 for some I satisfying 3 :1...