(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.

(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.

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