A Euclidean Ramsey Problem
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- Exoo Discrete Comput Geom (2003) 29: 223. doi:10.1007/s00454-002-0780-5
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Abstract. Let Cn denote the set of points in Rn whose coordinates are all 0 or 1 , i.e., the vertex set of the unit n -cube. Graham and Rothschild  proved that there exists an integer N such that for n ≥ N , any 2-coloring of the edges of the complete graph on Cn contains a monochromatic plane K4 . 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.