Some Topics On Ramsey Numbers And Euclidean Ramsey Problems

Ramsey theory is a branch of discrete mathematics,which has wide applications in Theoretical Computer Science,Information Theory,Decision Science and other fields.It reveals that complete disorder is impossible,and orderly dialectical unity must appear in disorder.The Ramsey number of graph is one of the parameters in the study of Ramsey theory.It has great theoretical significance to solve the exact values or upper and lower bounds of the Ramsey number of graph.Research on Ramsey numbers began in 1930,when Ramsey proposed the Ramsey problem.Given a graph G,and a positive integer k,define the Ramsey number R_k（G）as the minimum number of vertices N such that any k-edge-coloring of K_n（n≥ N）contains a monochromatic copy of G.Much like the definition of Ramsey number,given two graphs G and H,and a positive integer k,define the Gallai-Ramsey number gr_k（G:H）as the minimum number of vertices N such that any k-edge-coloring of K_n（n≥ N）contains either a rainbow copy of G or a monochromatic copy of H.The Euclidean Ramsey problem arises when we consider the monochromatic structure occurring under the coloring of the points in Euclidean space.A finite set of points is called a configuration in Euclidean space,configuration X is said to be rRamsey if we arbitrarily r-partite the points in Euclidean space E^N,that is to say E^N=C₁ ∪…∪ C_r,there will always exist some C_i contain a configuration that congruent with X,we write as E^N（?）Cong（X）.Configuration[X;Y]is said to be r-Gallai-Ramsey if we arbitrarily r-partite the points in Euclidean space E^N,that is to say E^N=C₁ ∪…U C_r,there will always exist two sets X,Y （?）E^N satisfied|X ∩ C_i|=1 for any i∈｛1,2,…,r} or there are some C_j contain a configuration that congruent with Y,which can be written as E^N（?）Cong[X;Y].Let graph H be a union graph of some star graphs and path graphs,graph G be a K_1,3,4-path P₅ or P₄⁺.In this paper,we determined the exact values or upper and lower bounds of Ramsey numbers R₂（H）and R₃（H）in Chapter 2.The exact values or upper and lower bounds of Gallai-Ramsey number gr_k（G:H）have been given in Chapter 3.In Chapter 4,we extended some Euclidean Ramsey theorems of Erdos et al.and gave some results of Euclidean Gallai-Ramsey theory involving simplex or isosceles triangle.