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Graham's number is a giant number that is the upper limit for solving a certain problem in Ramsey's theory. It is some very large power of the triple, which is written using Knuth notation. Named after Ronald Graham. It became known to the general public after Martin Gardner described it in his column

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 " width="135" height="240">
Graham's number is a giant number that is the upper limit for solving a certain problem in Ramsey's theory. It is some very large power of the triple, which is written using Knuth notation. Named after Ronald Graham. It became known to the general public after Martin Gardner described it in his column "Mathematical Games" in Scientific American in November 1977, where it was said: "In an unpublished proof, Graham recently set a boundary so large that it holds the record as the largest number ever used in a serious mathematical proof." In 1980, the Guinness Book of World Records echoed Gardner's claims, further fueling public interest in this number. The Graham number is an unimaginable number of times larger than other well-known large numbers such as the googol, the googolplex, and even larger than the Skews number and the Moser number. The entire observable universe is too small to accommodate the ordinary decimal notation of the Graham number (it is assumed that the notation of each digit occupies at least the Planck volume). Even the power towers of the view a b c ÷ ÷ ÷ {\ displaystyle a ^ {b ^ {c ^ {\ cdot ^ {\ cdot ^ {\ cdot}}}}}} are useless for this purpose (in the same sense), although this number can be written using recursive formulas such as Knuth's notation or equivalent, which was done by Graham. The last 500 digits of Graham's number are ...02425950695064738395657479136519351798334535362521 43003540126026771622672160419810652263169355188780 38814483140652526168785095552646051071172000997092 91249544378887496062882911725063001303622931916080 25459461494578871427832350829242102091825896753560 43086993801689249889268099510169055919951195027887 17830837018340236474548882222161573228010132974509 27344594504343300901096928025352751833289884461508 94042482650181938515625357963996189939679054966380 03222348723967018485186439059104575627262464195387. In modern mathematical proofs, numbers even larger than Graham's number are sometimes found, for example, in the work with Friedmann's finite form in Kraskal's theorem - the so-called TREE (3). Content 1 Graham's problem 2 Determination of the Graham number 2.1 Scale of Graham number 3 See also 4 Literature 5 References Graham problem Example: 2 colors and a 3-dimensional cube containing one single-color 4-vertex coplanar complete subgraph. The subgraph is shown below the cube. This cube will not contain such a subgraph if, for example, the bottom edge of the present subgraph is replaced by blue - which proves by a counterexample that N * > 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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