Colour small squares on the standard squared paper in two colors A, B. Name two small squares with common side as "neighbor". Let every colored set be "path connected": for any two small squares of the color A(resp. B) there is a sequence of color A(resp. B) neighbor squares from one to another. Could you help me to prove that must there exists square $3\times 3$ which has 6 squares of same color?

(It is clear that there is infinite path of each color... I constructed a lot of finite examples without desired square $3\times 3$, but they haven't common structure...)

Question: does this $3\times 3$ square exists? I think, yes, but I can't prove it.

Added: There is a counterexample, two spirals without desired square $3\times 3$.