infinite one-color subset of lattice points, symmetric wrt a point, all 2 colors

Art of Problem Solving

AoPS Online

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

w

17 users

TOPICS

No tags match your search

M

No tags match your search

M

infinite one-color subset of lattice points, symmetric wrt a point, all 2 colors

G H J

Reveal topic tags

combinatorial geometry

central symmetry

Kharkiv Masters

L

G H BBookmark kLocked kLocked NReply

Source: 2018 1st Kharkiv Masters Tournament X-XI Individual p6

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

30629 posts

#1 h May 21, 2022, 6:56 PM

Y by

Points with integer coordinates on the Cartesian plane were painted in two colors. Prove that there exists an infinite one-color subset symmetric with respect to some point of the plane.

This post has been edited 2 times. Last edited by parmenides51, May 21, 2022, 7:22 PM

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

2459 posts

#2 h May 23, 2022, 1:47 PM • 2 Y

Y by Iceternity, parmenides51

Suppose, for the sake of contradiction, the claim is not true. Consider the points $x_1=(1,1),x_2=(-1,1),x_3=(-1,-1,),x_4=(1,-1)$ . For any lattice point $x$ with sufficiently large $\|x\|$ , the point symmetrical to $x$ with respect to $x_i,i=1,2,3,4$ is colored in the opposite color of $x$ . Indeed, if it fails, we would construct an infinite monochromatic set of lattice points central symmetric with respect to some point among $x_i$ .
Further, take a point $x$ with $\|x\|$ sufficiently large and consider the point $x'$ symmetrical to $x$ with respect to $x_i$ and the point $x''$ - symmetrical to $x$ with respect to $x_{i+1}$ . They must be colored with the same color. Choosing appropriately $x,x_i,x_{i+1}$ we get that the points $(N,N), (N-2,N),(N-4,N),\dots, (-N,N)$ are colored with the same color, where $N$ is sufficiently large. Then, the same for the points $(-N,N),(-N,N-2)\dots, (-N,-N)$ . Hence, $(N,N)$ and $(-N,-N)$ have the same color, when $N$ is large enough. It means there are infinitely many lattice points central symmetric with respect to the origin, and colored in the same color, contradiction.

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a