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Tuesday 24 March 2026 07:17:34 GMT
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3. Graham's number is related to the following problem in Ramsey's theory: Consider n {\ displaystyle n} -dimensional hypercube and connect all pairs of vertices to obtain a complete graph with 2 n {\ displaystyle 2 ^ {n}} vertices. Paint each edge of this graph either red or blue. At what is the smallest value n {\ displaystyle n} does each such coloring necessarily contain a single-color complete subgraph with four vertices, all of which lie in the same plane? Graham and Rothschild proved in 1971 that this problem has a solution. N ∗ {\ displaystyle N ^ {*}}, and showed that 6 ≤ N ∗ ≤ N {\ displaystyle 6\ leqslant N ^ {*}\ leqslant N}, where N {\ displaystyle N} is a specific, precisely defined, very large number. In Knuth's arrow notation language, it can be written as N = F 7 ( 12 ) {\ displaystyle N = F ^ {7} (12)}, where F ( n ) = 2 ↑ n 3 {\ displaystyle F (n) = 2\ uparrow ^ {n} 3}. This number is referred to as the "Little Graham number." The lower bound was improved by Exu in 2003 and Barkley in 2008, which showed that N ∗ {\ displaystyle N ^ {*}} should be at least 13. Then the upper limit was improved to 2 ↑ 3 6 {\ displaystyle 2\ uparrow ^ {3} 6} and then up to 2 ↑ ↑ 2 ↑ ↑ 2 ↑ ↑ 9 {\ displaystyle 2\ uparrow\ uparrow 2\ uparrow 2\ uparrow\ uparrow 9}. Thus, 13 ≤ N ∗ ≤ 2 ↑ ↑ 2 ↑ ↑ 2 ↑ ↑ 9 {\ displaystyle 13\ leqslant N ^ {*}\ leqslant 2\ uparrow\ #fyp #based #larping #israel "
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