A Euclidean Ramsey Problem

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 [2] 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.