| In 1930,Frank Ramsey,the University of Cambridge economic mathemati-cian,proved a theorem in his paper "On a problem of formal logic",which now is known as the Ramsey theorem.Ramsey theory is stated as follows:for any graph H,there exists an integer N such that any red/blue edge-coloring of the complete grapbKh contains a,monochrormatic siubgraph H.Ramsey theory is an important branch of modern graph theory,whose study has deep influence to the development of graph theory.The problems on bipartite Ramsey numbers have attracted a lot of attention on graph Ramsey theory.In this thesis,we mainly study several bipartite Ramsey numbers.Let Hi and H2 be bipartite graphs,the bipartite Ramsey number br(H1;H2)is defined to be the minimum integer N such that any red/blue edge-coloring of the complete bipartite graph KN,N contains either a red copy of H1 or a blue copy of H2.In Chapter 1,we mainly introduce the history of Ramsey theory and some related results.In particular,we focus on classical Ramsey numbers and bipartite Rarnsey numbers.In Chapter 2,we mainly study the multicolor bipartite Ramsey numbers.It is shown that for any fixed positive integers t ≥2 and s>(t+ 1)+ 1,there exists a constant c = c(t)>0 such that br2(Kt,s,Kn,n)>c(n log log n/log2 n)t for sufficiently large n.And for k ≥ 3,we get brk(Ktt,;Kn,n)= θ(n/logn)t for sufficiently large n.Furthermore,we obtain a general result as follows.Let t ≥2 and k>3 be fixed integers,If H is a bipartite graph with Turan number e:x(N,H)= O(N2-1/t),then br2(H;-Kn,n)≥c(nloglogn/2nloglogn)t,and brk(H;Kn,n)=θ(nt/logtn),where c = c(t,k)>0.In Chapter 3,we introduce the bipartite Ramsey numbers of the cycles.Regularity lemma is an important tool of extremal graph.And regularity lemma has produced profound influence to extremal graph.We use regularity lemma to obtain the bound of br(C2;C2n).It is shown that br(C2n;C2n)=(2 + 0(1))n,for sufficiently large n. |
