# $\kappa$-coloring of $\mathbb{R}^2$ and triangle with area 1

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

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 $a$.
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 $0.5 < 1$.