关于Schur数的两个不等式

Two Inequalities on Schur Numbers

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.

For every integer k ≥ 1, one only can find finite integers such that： there exists a k-coloring of the set [ 1, n ] such that there isn＇ t any monochromatic solution to in [ 1, n ] 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.