logical deduction - Colouring the reals

If every real number is coloured either black or white, prove that there exist distinct real numbers $a,b,c$ all of the same colour such that $a-b=b-c$.

Answer

WLOG, 2 and 4 are both black. (This is WLOG because we can scale and translate the real line freely, and can also swap the colors.)

If 0 is black, then

$(a, b, c) = (4, 2, 0)$ are all black.
This satisfies $4-2=2-0$.

If 3 is black, then

$(a, b, c) = (4, 3, 2)$ are all black.
This satisfies $4-3=3-2$.

If 6 is black, then

$(a, b, c) = (6, 4, 2)$ are all black.
This satisfies $6-4=4-2$.

If none of ${0,3,6}$ is black, then

$(a, b, c) = (6, 3, 0)$ are all white.
This satisfies $6-3=3-0$.