关于Ramsey数r_n及Schur数s_n的上界

ON UPPER BOUNDS OF RAMSEY NUMBER r_n AND SCHUR NUMBER s_n

本文主要讨论Ramsey数及Scbur数.着重讨论如何改进他们的上界.文中应用了初等数论.级数并结合组合论的方法,反复应用整数的奇、偶性及鸽笼原理,从而大大降低了Ramsey数及Schur数上界.即对任意顶点个数不小于n!(3/2+sh1)+1的完全图的任一n边着色,一定有一个同色三角形.

This paper discusses the Ramsey Number and Schur Number. The main purpose is to improve the upper bounds of the Ramsey Number rn and the Schur Number sn. Through a combination of methods used in number theory,series and combinatorial theory,the upper bounds of the Ramsey Number rn and the Schur Number sn are largely improved by repeatedly using pigeonhole principle and the evenness of the integer numbers.