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What is the largest cardinal number integer $\kappa$ such that every $\kappa$-coloring of $\mathbb{R}^2$ contains a triangle with area 1 and all vertices of the same color?

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I think the question would be more well-posed if you instead asked for the least $\kappa$ for which the property holds for all cardinals less than $\kappa$. –  Kevin Ventullo Nov 30 '11 at 2:39
    
Did you mean "largest"? –  Vladimir Reshetnikov Nov 30 '11 at 2:55
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up vote 10 down vote accepted

R. Graham in "On Partitions of $\mathbb E^n$" , J. Combinatorial Theory Ser. A, 28 (1980), 89–97, proves that for any finite coloring of $\mathbb R^n$ and any positive $a\in \mathbb R$ there is a monochromatic simplex of volume $a$.

The question is originally due to Gurevich from the 70's. Graham's proof has been simplified, see for example the paper "Monochromatic simplices of any volume" by A. Dumitrescu, M. Jiang, Discrete Mathematics, Vol. 310, 4, 2010, 956-960.

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I think that you can always find such a triangle for any finite $\kappa$. Not sure how to prove that though.

There are countably infinite colorings that do not contain such a triangle. Partition the plane into a checkerboard of unit squares and color each square a different color. Then, the largest monochromatic triangle has area $0.5 < 1$.

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