A Euclidean Ramsey Problem
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  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.