# Show circle with points coloured red and blue must have monochromatic red equilateral triangle

Colour each point on a circle of radius $$\frac{1}{2}$$ red or blue, such that the region of blue points has length $$1$$. Prove that we can inscribe an equilateral triangle in the circle such that all three vertices are red.

I think the Pigeonhole Principle will be involved, but don't quite see how to apply it. The length condition also seems a bit hard to work with, so any hints or suggestions would be much appreciated.

• What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable? – John Hughes Dec 9 '18 at 21:41
• Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length. – Prasiortle Dec 9 '18 at 21:53

Make all the red points that are a distance exactly $$\frac {2\pi}3$$ away from a blue point blue. The measure of the blue points is now no more than $$3$$, but the circumference of the circle is $$\pi$$. There is at least $$\pi-3$$ of the circle still colored red and any of the red points is on an all red equilateral triangle.