• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.1001-1011
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• 2018

We introduce some variants of the finite Ramsey theorem. The variants are based on a refinement of homogeneity. In particular, they cover homogeneity, minimal homogeneity, end-homogeneity as special cases. We also show how to obtain upper bounds for the corresponding Ramsey numbers.

• Journal of applied mathematics & informatics
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• v.27 no.3_4
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• pp.909-913
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• 2009

For a family of graphs $\mathcal{H}$ and an integer k, we denote by $R^k(\mathcal{H})$ the corresponding k-Ramsey number, which is defined to be the smallest integer n such that every k-coloring of the edges of $K_n$ contains a monochromatic copy of a graph in $\mathcal{H}$. The local k-Ramsey number $R^k_{loc}(\mathcal{H})$ and the mean k-Ramsey number $R^k_{mean}(\mathcal{H})$ are defined analogously. Let $\mathcal{G}$ be the family of non-bipartite graphs and $T_n$ be the family of all trees on n vertices. In this paper we prove that $R^k_{loc}(\mathcal{G})=R^k_{mean}(\mathcal{G})$, and $R^2(T_n)$ < $R^2_{loc}(T_n)4 = $R^2_{mean}(T_n)$ for all $n\;{\ge}\;3$.

• Journal of the Korea Society of Computer and Information
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• v.20 no.3
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• pp.69-74
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• 2015

This paper offers solutions to unresolved $43{\leq}R(5,5){\leq}49$ and $102{\leq}R(6,6){\leq}165$ problems of Ramsey's number. The Ramsey's number R(s,t) of a complete graph $k_n$ dictates that n-1 number of incidental edges of a arbitrary vertex ${\upsilon}$ is dichotomized into two colors: (n-1)/2=R and (n-1)/2=B. Therefore, if one introduces the concept of distance to the vertex ${\upsilon}$, one may construct a partite graph $K_n=K_L+{\upsilon}+K_R$, to satisfy (n-1)/2=R of {$K_L,{\upsilon}$} and (n-1)/2=B of {${\upsilon},K_R$}. Subsequently, given that $K_L$ forms the color R of $K_{s-1)$, $K_S$ is attainable. Likewise, given that $K_R$ forms the color B of $K_{t-1}$, $K_t$ is obtained. By following the above-mentioned steps, $R(s,t)=K_n$ was obtained, satisfying necessary and sufficient conditions where, for $K_L$ and $K_R$, the maximum distance should be even and incidental edges of all vertices should be equal are satisfied. This paper accordingly proves R(5,5)=43 and R(6,6)=91.