# A Euclidean Ramsey Problem

## Authors

DOI: 10.1007/s00454-002-0780-5

- Cite this article as:
- Exoo Discrete Comput Geom (2003) 29: 223. doi:10.1007/s00454-002-0780-5

**Abstract. ** Let *C*_{
n
} denote the set of points in *R*^{
n
} whose coordinates are all *0* or *1* , i.e., the vertex set of the unit *n* -cube. Graham and Rothschild [2] proved that there exists an integer *N* such that for *n ≥ N* , any 2-coloring of the edges of the complete graph on *C*_{
n
} contains a monochromatic plane *K*_{
4
} . Let *N*^{
*
} be the minimum such *N* . They noted that *N*^{
*
} must be at least *6* . Their upper bound on *N*^{
*
} has come to be known as *Graham's number* , often cited as the largest number that has ever been put to any practical use. In this note we show that *N*^{
*
} must be at least *11* and provide some experimental evidence suggesting that *N*^{
*
} is larger still.