For a graph $H$ and an integer $k\ge1$, the $k$-color Ramsey number $R_k(H)$ is least $N$ such that every $k$-coloring of edges complete $K_N$ contains monochromatic copy $H$. Let $C_m$ denote cycle on $m\ge4$ vertices let $\Theta_m$ family graphs obtained from by adding additional edge joining two non-consecutive vertices. Unlike odd cycles, little known about general behavior $R_k(C_{2n})$ ex...

We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assump...

1 The Ramsey number R(G1,G2) is the smallest integer p such that for any graph G on p vertices 2 either G contains G1 or G contains G2, where G denotes the complement of G. In this paper, some 3 new bounds with two parameters for the Ramsey number R(G1,G2), under some assumptions, are 4 obtained. Especially, we prove that R(K6 − e, K6) ≤ 116 and R(K6 − e, K7) ≤ 202, these improve 5 the two uppe...

Graham, Jan Karel Lenstra, and Robert E. Tarjan Discrete.This article is about the large number named after Ronald Graham. While Graham was trying to explain a result in Ramsey theory which he had derived with his collaborator Bruce Lee. Ramseys Theorem for n-Parameter Sets PDF.on Ramsey theory. Szemerédis most famous theorem is at the heart of Ramsey theory. Graham is the informal administrato...

The study of problems concerning subdivisions graphs has a rich history in extremal combinatorics. Confirming conjecture Burr and Erd?s, Alon proved 1994 that subdivided have linear Ramsey numbers. Later, Alon, Krivelevich Sudakov showed every n-vertex graph with at least ?n2 edges contains 1-subdivision the complete on c?n vertices, resolving another old Erd?s. In this paper we consider direct...

We make use of a particular linear spatial embedding, the cyclic polytope, in an exploration of bounds on the Ramsey number of knots. Using arc presentations to simplify knotted cycles of this embedding, we examine the relationships between the Ramsey number, bridge number, crossing number, stick number and arc index of knots. In particular we show the Ramsey number is at least as large as the ...