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?



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


