• 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.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 1990.04a
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• pp.163-170
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• 1990

Clean rooms are widely used in high technology industries. Currently within the microelectronics industry there is an explosive growth in the number of clean rooms. Therefore, special consideration of clean room workers is needed to the work induced stresses from contamination avoidance, clothing requirements, and confinements [1].

• Korean Journal of Mathematics
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• v.28 no.2
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• pp.379-389
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• 2020

An r-color Rado number N = R(𝓛, r) for a system 𝓛 of equations is the least integer, provided it exists, such that for every r-coloring of the set {1, 2, …, N}, there is a monochromatic solution to 𝓛. In this paper, we study the 2-color Rado number R(𝓔, 2) for 𝓔 : x₁ + x₂ + ⋯ + x_n = y₁ + y₂ = z when n ≥ 4.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1341-1354
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• 2021

Let r ≥ 2, and let k_i ≥ 2 for 1 ≤ i ≤ r. Mixed van der Waerden's theorem states that there exists a least positive integer w = w(k₁, k₂, k₃, …, k_r; r) such that for any n ≥ w, every r-colouring of [1, n] admits a k_i-term arithmetic progression with colour i for some i ∈ [1, r]. For k ≥ 3 and r ≥ 2, the mixed van der Waerden number w(k, 2, 2, …, 2; r) is denoted by w₂(k; r). B. Landman and A. Robertson [9] showed that for k < r < $\frac{3}{2}$(k - 1) and r ≥ 2k + 2, the inequality w₂(k; r) ≤ r(k - 1) holds. In this note, we establish some results on w₂(k; r) for 2 ≤ r ≤ k.

• Economic Analysis
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• v.25 no.2
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• pp.1-53
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• 2019

This paper examines monetary policy quantitatively in a two-agent and small-scale New-Keynesian economy with debt-constrained households that cannot smooth their consumption intertemporally and frictionlessly since highly indebted households are not allowed to borrow above a certain debt ceiling in incomplete financial markets without additional risk premiums due to information asymmetry between savers and borrowers. We find that, in the event of cost shocks, the asymmetric responses of borrowing households without, and saving households with, dividend incomes lead to different labor supplies and consumptions over heterogeneous households, and eventually to an extension of the monetary policy transmission channels. The income effect and low elasticity of the labor supply play key roles in such asymmetric responses over heterogeneous households. We also find that the social welfare in a flexible inflation targeting (FIT) monetary policy, in which both the inflation gap and the output gap are considered in an integrated manner when policy-making, is similar to that of the Ramsey optimal monetary policy (ROP), in which the shares of debt-constrained households, as well as all economic states, including both the inflation gap and output gap, are considered comprehensively for policy-making, and that it is greater than that of simple inflation targeting (SIT) monetary policy, in which only the inflation gap is considered mechanically for policy-making. Such social welfare implies that a FIT policy may still work even in an economy with a sizable number of debt-constrained households. Further, the responses of cost shocks to consumption and labor supply are dying out more slowly under FIT and ROP policies than under an SIT policy.