分析方法在Ramsey数估值中的应用

“Yusheng等人曾给出-个独立数的下界公式：α（G）≥Nfa＋1（d），其中fa（x）=∫0^1（1-t）1/a dt/（a＋（x-a）·t）。为了得到r（H，Kn）的上界，可以考虑建立不合H作为子图的临界图G的独立数的下界。即通过对临界图G及其邻域导出子图e的平均次数的分析，得出G的阶（顶点数）Ⅳ与n之间的不等式关系。再利用函数五（x）的分析性质得出当n趋于无穷大时，N＋1的最小可能渐近表达式，即为r（H，Kn）的渐近上界。主要介绍这种分析方法在解决Kk＋Kt，“K1＋Cm”，“Km．t”等图形和完全图Ramsey数渐近上界问题中的应用。

The Application of Analytic Methods on Estimating the Asymptotic Behavior of Graph Ramsey Function

Abstract： Yusheng Li, et al have given α（G）≥Nfa＋1 （ d）, where fa（x）=∫0^1（1-t）1/a dt/（a＋（x-a）·t） To abtain an upper bound of r（H,Kn）, one may turn to establish a lower bound of the independence number of any H - free graph with a fixed number of vertices. That is to say, we can get the ine- quality between n and N（ the order of critical graph G） by means of reckoning the average degree of G and the subgraph of G induced by the neiborhood of some one vertex v. Then utilizing the analytic proper- ties of function fa （X） and analytic operation we can achieve the asymptotic expression of N ＋ 1, namely the asymptotic upper bounds of r（H,Kn）. This paper surveys some main results and analytic methods on estimating the asymptotic behavior of graph Ramsey numbers, including Kk ＋ Kl, K1 ＋ Cm, Km.k versus complete graphs Ramsey numbers.