I agree with Rainman.
The proper version comes from "What is the Name of this Book" by Raymond Smullyan and is called 'The Drinking Principle'.
Smullyan also states a dual paradox:
There is at least one person such that if anybody drinks, then he does,
and more dramatic version of TDP: There is a woman on earth such that if she becomes sterile, the whole human race will die out.

(1a) Paint the plane using 3 colors in such way that no 3-colored straight line exists.
(1b) Prove that if plane is painted with 4 colors, then a 3-colored straight line exists.
(2a) Paint the plane using as many colors as you can in such way that no 4-colored straight line exists.
(2b) Let m be the number of colors you used in (2a). Prove that if plane is painted with (m+1) colors, then a 4-colored straight line exists.
(3) What about 5-colored straight lines?
(4) What about n-colored straight lines?
Note: I don't have solutions to all of the problems above. Partial solutions are welcome.

If I understand correctly, the following pattern is not considered "crossing":
__Ω_Ω_Ω_Ω_Ω__
The loops are closed, but it is "touching" not "crossing".