Guantao Chen Department of Mathematics and Computing Science Georgia State University ATLANTA GA 30303 USA Given a graph theoretic parameter f, a graph H and a positive integer m, the mixed ramsey number f(m,H) is defmed as the least positive integer p such that for any graph G of order p either f(G) ~ m or G contains H as a subgraph. In this paper we determine the mixed ramsey number X2(m,K( I...

We prove that the chromatic Ramsey number of every odd wheel W2k+1, k ≥ 2 is 14. That is, for every odd wheel W2k+1, there exists a 14-chromatic graph F such that when the edges of F are two-coloured, there is a monochromatic copy of W2k+1 in F , and no graph F with chromatic number 13 has the same property. We ask whether a natural extension of odd wheels to the family of generalized Mycielski...

Recently, Souza introduced blowup Ramsey numbers as a generalization of bipartite numbers. For graphs $G$ and $H$, say $G\overset{r}{\longrightarrow} H$ if every $r$-edge-coloring contains monochromatic copy $H$. Let $H[t]$ denote the $t$-blowup Then number $G,H,r,$ $t$ is defined minimum $n$ such that $G[n] \overset{r}{\longrightarrow} H[t]$. proved upper lower bounds on are exponential in $t$...

For a graph class G and any two positive integers i and j, the Ramsey number RG(i, j) is the smallest positive integer such that every graph in G on at least RG(i, j) vertices has a clique of size i or an independent set of size j. For the class of all graphs, Ramsey numbers are notoriously hard to determine, and they are known only for very small values of i and j. Even if we restrict G to be ...

It is shown in this paper that the pair (G, H) is Ramsey infinite when both G and H are forests, with at least one of G or H having a non-star component. In addition, an infinite subfamily of R(PP.) is constructed .

The Ramsey number r(F, G) is the least number N such that in every two-coloring (R, B)=(red, blue) of the edges of KN , either there is a red copy of F or else a blue copy of G . The inequality (1) holds in view of the fact that the edges of the complete graph of order (m-1) (n-1) +s-1 can be given the coloring in which the blue graph, denoted (B), is isomorphic to (m-1)K„_~UKS_~ . Then the red...

The Ramsey number of a graph G is the least number t for which it is true that whenever the edges of the complete graph on t vertices are colored in an arbitrary fashion using two colors, say red and blue, then it is always the case that either the red subgraph contains G or the blue subgraph contains G. A conjecture of P. Erdos and S. Burr is settled in the afftrmative by proving that for each...

On-line Ramsey theory studies a graph-building game between two players. The player called Builder builds edges one at a time, and the player called Painter paints each new edge red or blue after it is built. The graph constructed is called the background graph. Builder’s goal is to cause the background graph to contain a monochromatic copy of a given goal graph, and Painter’s goal is to preven...

We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on 2n vertices are bounded by O(n3) and O(n10), in the convex and general case, respectively. We then apply similar methods to prove an nO(log(n)) upper bound on the Ramsey number of a path wit...