关于Schur数的两个不等式

对每个整数k≥1,仅有有限个整数n满足:存在整数集合[1,n]上的一种k着色,使x+y=z的单色解在[1,n]内不存在.这些数最大的叫作Schur数,记为S(k).如果把条件加强为数组(x,y,z)中各数互不相同,满足条件的数S*(k)称为强Schur数.本文给出了关于这两种Schur数的两个不等式,并且给出了强Schur数的新下界.

Two Inequalities on Schur Numbers

For every integer k≥1,one only can find finite integers such that: there exists a k-coloring of the set such that there isn't any monochromatic solution to in x+y=z.The maximum integers satisfying such condition are called Schur numbers and denoted by S(k).If we restrict to triplets(x,y,z)of pairwise distinct integers,the integers S~(*)(k) called strong Schur numbers.The purpose of this paper is to give two inequalities on two kinds of Schur numbers and we also obtain a new lower bound of S~(*)(k) by the inequalities.