My friend Cho Seung Hee gives 32 hugs Mathematical Games by Martin Gardner Scientific American, November 1977 In an unpublished proof, Graham recently established a bound so vast that it holds the record for the largest number ever used in a serious mathematical proof. The problem that Graham solved belongs to a branch of combinatorics called Ramsey theory, named for the British mathematician Frank Plumpton Ramsey. The field is concerned with properties that are preserved under partitioning. A typical Ramsey problem is to find the minimum number of elements a structure must have to ensure that a certain property holds, regardless of how the structure is divided into parts. Graham’s problem can be stated in terms of an N-dimensional hypercube. Imagine a cube in three dimensions. It has 8 vertices and 12 edges. Now consider a hypercube of N dimensions, which has \(2^{N}\) vertices. Connect every pair of vertices by a straight line. This forms a complete graph with \(2^{N}\) vertices, containing a total of \(2^{N-1}(2^N - 1)\) lines. Now suppose we color every single line in this graph using only two colors: red and blue. We want to know how large N must be before we can guarantee that no matter how the coloring is done, there will always exist four coplanar vertices (vertices that lie on a single flat plane) such that all six lines connecting them are of the same color. Graham, working with Bruce L. Rothschild, proved in 1971 that such a number \(N^{*}\) does exist. However, they could not determine its exact value. They could only establish an upper bound for \(N^{*}\). To express this upper bound, ordinary exponential notation is completely useless. Even the familiar tower of exponents: \(3^{3^{3^{\cdot ^{\cdot ^{\cdot ^{3}}}}}}\) cannot be written down if the tower has more than a few steps, because the number of digits would exceed the number of atoms in the observable universe. To overcome this, Graham used a new arrow notation developed by Donald Knuth. A single arrow \(\uparrow \) signifies ordinary exponentiation. Thus: \(3\uparrow 3=3^{3}=27\) Two arrows \(\uparrow\uparrow\) indicate a tower of exponents: \(3\uparrow \uparrow 3=3\uparrow (3\uparrow 3)=3^{3^{3}}=3^{27}=7,625,597,484,987\) Three arrows \(\uparrow\uparrow\uparrow\) represent a tower of towers, and four arrows \(\uparrow\uparrow\uparrow\uparrow\) represent a tower of towers of towers. Graham began his construction with a number he called g₁: \(g_{1}=3\uparrow \uparrow \uparrow \uparrow 3\) This number g₁ is already so immensely large that it is completely beyond human comprehension. Yet, it is only the first step. Graham then constructed a sequence of 64 numbers, where the number of arrows in each step is determined by the value of the preceding number: \(g_{2}=3\uparrow ^{g_{1}}3\) \(g_{3}=3\uparrow ^{g_{2}}3\) and so on, until he reached: \(g_{64}=3\uparrow ^{g_{63}}3\) This final value, g₆₄, is the upper bound established by Graham. It is a number so unimaginably vast that its digits cannot be written down, nor can they be contained within the boundaries of the known physical universe. It stands as a monument to the staggering scale of combinatorial possibilities. #ez #fyp #xyzbca #rampage #viral #рекомендации - @zarxzeon51