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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.

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 $3\times 3$.

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?

(I can't prove it. 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...)

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.

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?

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

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?

(I can't prove it. 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...)