The Ramsey number R(G,H) has been actively studied for the past 40 years, and it was determined for a large family of pairs (G,H) of graphs. The Ramsey number of paths was determined very early on, but surprisingly very little is known about the Ramsey number for the powers of paths. The r-th power P r n of a path on n vertices is obtained by joining any two vertices with distance at most r. We...

The graph Ramsey number R(G,H) is the smallest integer r such that every 2-coloring of the edges of Kr contains either a red copy of G or a blue copy of H . We find the largest star that can be removed from Kr such that the underlying graph is still forced to have a red G or a blue H . Thus, we introduce the star-critical Ramsey number r∗(G,H) as the smallest integer k such that every 2-colorin...

Bounds are determined for the smallest m = Dr(Kn) such that every drawing of Km in the plane (two edges have at most one point in common) contains at least one drawing of Kn with the maximum number (:) of crossings. For n = 5 these bounds are improved to 11 :::; Dr(K5) 113. A drawing D( G) of a graph G is a special realization of G in the plane. The vertices are mapped into different points of ...

A subgraph of an edge-colored graph is called rainbow if all of its edges have different colors. For a graph G and a positive integer n, the anti-Ramsey number ar(n, G) is the maximum number of colors in an edge-coloring of Kn with no rainbow copy of H. Anti-Ramsey numbers were introduced by Erdős, Simonovits and Sós and studied in numerous papers. Let G be a graph with anti-Ramsey number ar(n,...

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...

In [21], Frank Plumpton Ramsey proved what has turned out to be a remarkable and important theorem which is now known as Ramsey's theorem. This result is a generalization of the pigeonhole principle and can now be seen as part of a family of theorems of the same flavour. These Ramsey-type theorems all have the common feature that they assert, in some precise combinatorial way, that if we deal w...

Does the $n^{th}$ root of the diagonal Ramsey number converge to a finite limit? The answer is yes. A sequence can be shown to converge if it satifies convergence conditions other than or besides monotonicity. We show such a property holds for which the sequence of $n^{th}$ roots does converge, even if one has no a priori knowledge as to whether the sequence is monotone or not. We show also the...

Classical Multi-color Ramsey Theory pertains to the existence of monochromatic subsets of structured multicolored sets such that the subsets have a given property or structure. This paper examines a generalization of Ramsey theory that allows the subsets to have specified groupings of colors. By allowing more than one color in subsets, the corresponding minimal sets for finite cases tend to be ...