平面格点形心问题研究

The study on the centroid of the lattice point in the plane

在组合论和数论中 ,平面格点形心问题是对给定的自然数k,求这样的最小整数n(k) ,使得当n≥n(k)时 ,平面上任意几个格点中必存在k个格点的形心也是格点。显然n(1) =1,并容易求出n(2 ) =5。文献[1]用较复杂的组合设计方法确定出n(3) =9。本文提出一种简易的方法 ,给出n(3) =9的新证 ,并得到n(4)的改进上界。

The centroid of the lattice point in the plane is studied in this paper. Let n(k) be the smallest integer n such that, given any n latice points in the plane, some k of them have a lattice point centroid. Erickson gave a proof of n(3)=9 by the method of finite projective plane in [1]. Firstly, this paper gives a new and simple proof of n(3)=9 by geometric method and the pigeonhole principle. Secondly, it improves the upper bound of n(4) .