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 5 of 6: Iterate on nested infinite sets

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In plain words

Each round spends one of the finitely many colors, while the infinite subsequence leaves enough room for the next round.

x_{j_{k_1}+1,\,j_k-j_{k_1}}\in\mathcal C_{j_{k_1}+1},\quad C(x_{j_{k_1}+1,\,j_k-j_{k_1}})=\mathcal C_{j_k+1}

Detailed analysis

Choose the first index from the infinite set just obtained and inspect the corresponding row. Its infinitely many selected entries lie on circles already forbidden to $c_1$ , so an infinite pigeonhole subsequence has a new color $c_2$ . Repeating this construction produces colors $c_1,c_2,\ldots$ , each forbidden on the same type of infinitely many target circles; each new color is distinct because those circles already forbid all earlier colors.