平面格点形心问题研究

在组合论和数论中，平面格点形心问题是对给定的自然数κ，求这样的最小整数n(κ)，使得当n≥n(κ)时，平面上任意几个格点中必存在是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 study on the centroid of the lattice point in the plane

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) .