Problem 3

Given points

O

and

A

in the plane, every point is colored with one of a finite number of colors. For a point

X

, let

C(X)

be the circle centered at

O

with radius

OX+\frac{\angle AOX}{OX}

, where

\angle AOX

is measured in radians in

\left[0,2\pi\right)

. Prove that there is a point

X

, not on

OA

, such that the color of

X

appears on the circumference of

C(X)

Step 4 of 6: Use the first repeated color

Previous step Next step

In plain words

A repeated color on a source point bans that color from its target circle, otherwise the theorem would already be finished.

x_{1,j_k}\text{ all have color }c_1\ \Longrightarrow\ c_1\text{ is absent from }\mathcal C_{1+j_k}

Detailed analysis

Assume the desired point does not exist. Since the first row has finitely many colors, some color $c_1$ occurs at infinitely many entries $x_{1,j_k}$ . If $c_1$ occurred anywhere on $\mathcal C_{1+j_k}$ , that entry itself would be a desired point; hence $c_1$ is forbidden on every such circle.