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A square in the plane with 4 vertices of the same color

Puzzling Asked on March 30, 2021

Every point in the plane is colored either red or blue. Is it necessarily the case (i.e., is it true for all such colorings) that there exist some four points of the same color that are the vertices of a square?

3 Answers

I believe the answer is

Not a proof, but a heuristic argument:

EDIT: A "proof" by computer. At least this gives the evidence for the yes/no part, so that a more serious attempt can be made in the right direction.

Answered by Bubbler on March 30, 2021

Not an answer, but a suggestive result.

That doesn't prove that a square is inevitable, because it leaves open the possibility that

Answered by Rosie F on March 30, 2021

There is a relevant paper which I came across recently connected to this question:

  • "Extremal binary matrices without constant 2-squares" by Roland Bacher and Shalom Eliahou

In it, the authors prove that

which shows not only that

but also that

Further interesting points

Answered by hexomino on March 30, 2021

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