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...

In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, show that every sufficiently large graph with average degree $D$ and $n$ vertices each side has balanced independent set containing $(1-\epsilon) \frac{\log D}{D} n$ from for small $\epsilon > 0$. Secondly, prove the vertex maximum at most $\Delta$ can be partitioned into $(1+\epsilon)\frac{\Delta...

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...

Let H1, . . . ,Hk be graphs. The multicolor Ramsey number r(H1, . . . ,Hk) is the minimum integer r such that in every edge-coloring of Kr by k colors, there is a monochromatic copy of Hi in color i for some 1 ≤ i ≤ k. In this paper, we investigate the multicolor Ramsey number r(K2,t, . . . ,K2,t,Km), determining the asymptotic behavior up to a polylogarithmic factor for almost all ranges of t ...

Let f(n, p, q) be the minimum number of colors necessary to color the edges of Kn so that every Kp is at least q-colored. We improve current bounds on these nearly “anti-Ramsey” numbers, first studied by Erdős and Gyárfás. We show that f(n, 5, 9) ≥ 7 4 n− 3, slightly improving the bound of Axenovich. We make small improvements on bounds of Erdős and Gyárfás by showing 5 6 n+1 ≤ f(n, 4, 5) and f...