7
$\begingroup$

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?

$\endgroup$
2
  • $\begingroup$ 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 $\kappa$. $\endgroup$ Nov 30, 2011 at 2:39
  • $\begingroup$ Did you mean "largest"? $\endgroup$ Nov 30, 2011 at 2:55

2 Answers 2

10
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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 $0.5 < 1$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.