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| | '''Graham's number''' is a very, very big [[natural number]] that was defined by a man named Ronald Graham. Graham was solving a problem in an area of mathematics called [[Ramsey theory]]. He proved that the answer to his problem was smaller than Graham's number. |
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| |title = 组合数学 | |
| Combinatorics
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| | Graham's number is one of the biggest numbers ever used in a [[mathematical proof]]. Even if every digit in Graham's number were written in the tiniest writing possible, it would still be too big to fit in the [[observable universe]]. |
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| |header1 =Instructor
| | ==Context== |
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| |data2 = 尹一通
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| |label3 = Email
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| |data3 = yitong.yin@gmail.com yinyt@nju.edu.cn yinyt@lamda.nju.edu.cn
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| |data4= 计算机系 804
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| |header5 = Class
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| |label6 = Class meetings
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| |data6 = Thursday, 10am-12pm <br> 仙逸B-104
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| |label8 = Office hours
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| |data8 = Wednesday, 2-5pm <br>计算机系 804
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| |header9 = Textbook
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| |data10 = ''van Lint and Wilson,'' <br> A course in Combinatorics, 2nd Ed, <br> Cambridge Univ Press, 2001.
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| | Ramsey theory is an area of mathematics that asks questions like the following: |
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| This is the page for the class ''Combinatorics'' for the Fall 2011 semester. Students who take this class should check this page periodically for content updates and new announcements.
| | {{quote|<p>Suppose we draw some number of points, and connect every pair of points by a line. Some lines are blue and some are red. Can we always find 3 points for which the 3 lines connecting them are all the same color?}} |
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| = Announcement =
| | It turns out that for this simple problem, the answer is "yes" when we have 6 or more points, no matter how the lines are colored. But when we have 5 points or fewer, we can color the lines so that the answer is "no". |
| * <font size=3 color=red>第一、二次课的slides已发布，见lecture notes部分。</font>
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| = Course info =
| | Graham's number comes from a variation on this question. |
| * '''Instructor ''': 尹一通
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| :*email: yitong.yin@gmail.com, yinyt@nju.edu.cn,
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| :*office: 804
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| * '''Teaching fellow''': TBA
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| :*email: TBA
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| * '''Class meeting''': Thursday 10am-12pm, 仙逸B-104.
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| * '''Office hour''': Wednesday 2-5pm, 计算机系 804.
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| | {{quote|<p>Once again, say we have some points, but now they are the corners of an ''n''-dimensional [[hypercube]]. They are still all connected by blue and red lines. For any 4 points, there are 6 lines connecting them. Can we find 4 points that all lie on one [[Plane (mathematics)|plane]], and the 6 lines connecting them are all the same color?}} |
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| = Syllabus =
| | By asking that the 4 points lie on a plane, we have made the problem much harder. We would like to know: for what values of ''n'' is the answer "no" (for some way of coloring the lines), and for what values of ''n'' is it "yes" (for all ways of coloring the lines)? But this problem has not been completely solved yet. |
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| === 先修课程 Prerequisites === | | In 1971, Ronald Graham and B. L. Rothschild found a partial answer to this problem. They showed that for ''n''=6, the answer is "no". But when ''n'' is very large, as large as Graham's number or larger, the answer is "yes". |
| * 离散数学（Discrete Mathematics）
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| * 线性代数（Linear Algebra）
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| * 概率论（Probability Theory）
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| === Course materials ===
| | One of the reasons this partial answer is important is that it means that the answer is eventually "yes" for at least some large ''n''. Before 1971, we didn't know even that much. |
| * [[组合数学 (Fall 2011)/Course materials|教材和参考书清单]]
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| === 成绩 Grades === | | ==Definition== |
| * 课程成绩：本课程将会有六次作业和一次期末考试。最终成绩将由平时作业成绩和期末考试成绩综合得出。
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| * 迟交：如果有特殊的理由，无法按时完成作业，请提前联系授课老师，给出正当理由。否则迟交的作业将不被接受。
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| === <font color=red> 学术诚信 Academic Integrity </font>===
| | Graham's number is not only too big to write down all of its digits, it is too big even to write in [[scientific notation]]. In order to be able to write it down, we have to use [[Knuth's up-arrow notation]]. |
| 学术诚信是所有从事学术活动的学生和学者最基本的职业道德底线，本课程将不遗余力的维护学术诚信规范，违反这一底线的行为将不会被容忍。
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| 作业完成的原则：署你名字的工作必须由你完成。允许讨论，但作业必须独立完成，并在作业中列出所有参与讨论的人。不允许其他任何形式的合作——尤其是与已经完成作业的同学“讨论”。
| | We will write down a [[sequence]] of numbers that we will call '''g1''', '''g2''', '''g3''', and so on. Each one will be used in an equation to find the next. '''g64 is''' Graham's number. |
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| 本课程将对剽窃行为采取零容忍的态度。在完成作业过程中，对他人工作（出版物、互联网资料、其他人的作业等）直接的文本抄袭和对关键思想、关键元素的抄袭，按照 [http://www.acm.org/publications/policies/plagiarism_policy ACM Policy on Plagiarism]的解释，都将视为剽窃。剽窃者成绩将被取消。如果发现互相抄袭行为，<font color=red> 抄袭和被抄袭双方的成绩都将被取消</font>。因此请主动防止自己的作业被他人抄袭。
| | First, here are some examples of up-arrows: |
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| 学术诚信影响学生个人的品行，也关乎整个教育系统的正常运转。为了一点分数而做出学术不端的行为，不仅使自己沦为一个欺骗者，也使他人的诚实努力失去意义。让我们一起努力维护一个诚信的环境。
| | * <math>3\uparrow3</math> is 3x3x3 which equals 27. An arrow between two numbers just means the first number multiplied by itself the second number of times. |
| | * You can think of <math>3 \uparrow \uparrow 3</math> as <math>3 \uparrow (3 \uparrow 3)</math> because two arrows between numbers A and B just means A written down a B number of times with an arrow in between each A. Because we know what single arrows are, <math>3\uparrow(3\uparrow3)</math> is 3 multiplied by itself <math>3\uparrow3</math> times and we know <math>3\uparrow3 |
| | </math> is twenty-seven. So <math>3\uparrow\uparrow3</math> is 3x3x3x3x....x3x3, in total 27 times. That equals 7625597484987. |
| | * <math>3 \uparrow \uparrow \uparrow 3</math> is <math>3 \uparrow \uparrow (3 \uparrow \uparrow 3)</math> and we know <math>3\uparrow\uparrow3</math> is 7625597484987. So <math>3\uparrow\uparrow(3\uparrow\uparrow3)</math> is <math>3\uparrow \uparrow 7625597484987</math>. That can also be written as <math>3\uparrow(3\uparrow(3\uparrow(3\uparrow . . .(3\uparrow(3\uparrow(3\uparrow3)</math> with a total of 7625597484987 3s. This number is so huge, its digits, even written very small, could fill up the observable universe and beyond. |
| | ** Although this number may already be beyond comprehension, this is barely the start of this giant number. |
| | * The next step like this is <math>3 \uparrow \uparrow \uparrow \uparrow 3</math> or <math>3 \uparrow \uparrow \uparrow (3 \uparrow \uparrow \uparrow 3)</math>. This is the number we will call '''g1'''. |
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| = Assignments =
| | After that, '''g2''' is equal to <math>3\uparrow \uparrow \uparrow \uparrow \ldots \uparrow \uparrow \uparrow \uparrow 3</math>; the number of arrows in this number is '''g1'''. |
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| = Lecture Notes =
| | '''g3''' is equal to <math>3\uparrow \uparrow \uparrow \uparrow \uparrow \ldots \uparrow \uparrow \uparrow \uparrow \uparrow 3</math>, where the number of arrows is '''g2'''. |
| # [[组合数学 (Fall 2011)/Basic enumeration|Basic enumeration]] | [http://lamda.nju.edu.cn/yinyt/notes/comb2011/comb1.pdf slides1] | [http://lamda.nju.edu.cn/yinyt/notes/comb2011/comb2-1.pdf slides2]
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| # [[组合数学 (Fall 2011)/Generating functions|Generating functions]] | [http://lamda.nju.edu.cn/yinyt/notes/comb2011/comb2-2.pdf slides1]
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| # [[组合数学 (Fall 2011)/Sieve methods|Sieve methods]]
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| # [[组合数学 (Fall 2011)/Pólya's theory of counting|Pólya's theory of counting]]
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| # [[组合数学 (Fall 2011)/Counting and existence|Counting and existence]]
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| # [[组合数学 (Fall 2011)/Discrete probability|Discrete probability]]
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| # [[组合数学 (Fall 2011)/The probabilistic method|The probabilistic method]]
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| # [[组合数学 (Fall 2011)/Extremal graph theory| Extremal graph theory]]
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| # [[组合数学 (Fall 2011)/Matching theory|Matching theory]]
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| # [[组合数学 (Fall 2011)/Flow and matching | Flow and matching]]
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| # [[组合数学 (Fall 2011)/Optimization|Optimization]]
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| # [[组合数学 (Fall 2011)/Matroid|Matroid]]
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| # [[组合数学 (Fall 2011)/Extremal set theory|Extremal set theory]]
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| # [[组合数学 (Fall 2011)/Ramsey theory|Ramsey theory]]
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| # [[组合数学 (Fall 2011)/The Szemeredi regularity lemma|The Szemeredi regularity lemma]]
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| = Concepts =
| | We keep going in this way. We stop when we define '''g64''' to be <math>3\uparrow \uparrow \uparrow \uparrow \uparrow \ldots \uparrow \uparrow \uparrow \uparrow \uparrow 3</math>, where the number of arrows is '''g63'''. |
| * [http://en.wikipedia.org/wiki/Binomial_coefficient Binomial coefficient]
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| * [http://en.wikipedia.org/wiki/Composition_(number_theory) Composition of a number]
| | This is Graham's number. |
| * [http://en.wikipedia.org/wiki/Combination#Number_of_combinations_with_repetition Combinations with repetition], [http://en.wikipedia.org/wiki/Multiset_coefficient#Multiset_coefficients <math>k</math>-multisets on a set]
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| * [http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind Stirling number of the second kind]
| | ==Related pages== |
| * [http://en.wikipedia.org/wiki/Partition_(number_theory) Partition of a number]
| | * [[Knuth's up-arrow notation]] |
| * [http://en.wikipedia.org/wiki/Twelvefold_way The twelvefold way]
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| * [http://en.wikipedia.org/wiki/Partition_(number_theory)#Ferrers_diagram Ferrers diagram] (and the MathWorld [http://mathworld.wolfram.com/FerrersDiagram.html link])
| | [[Category:Mathematics]] |
| * [http://en.wikipedia.org/wiki/Inclusion-exclusion_principle The principle of inclusion-exclusion] (and more generally the [http://en.wikipedia.org/wiki/Sieve_theory sieve method])
| | [[Category:Hyperoperations]] |
| * [http://en.wikipedia.org/wiki/Derangement Derangement], and [http://en.wikipedia.org/wiki/M%C3%A9nage_problem Problème des ménages]
| | [[Category:Integers]] |
| * [http://en.wikipedia.org/wiki/Generating_function Generating function] and [http://en.wikipedia.org/wiki/Formal_power_series formal power series]
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| * [http://en.wikipedia.org/wiki/Fibonacci_number Fibonacci number]
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| * [http://en.wikipedia.org/wiki/Binomial_series Newton's formula]
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| * [http://en.wikipedia.org/wiki/Catalan_number Catalan number]
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| * [http://en.wikipedia.org/wiki/Double_counting_(proof_technique) Double counting] and the [http://en.wikipedia.org/wiki/Handshaking_lemma handshaking lemma]
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| * [http://en.wikipedia.org/wiki/Cayley's_formula Cayley's formula]
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| * [http://en.wikipedia.org/wiki/Pigeonhole_principle Pigeonhole principle]
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| :* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szekeres_theorem Erdős–Szekeres theorem]
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| :* [http://en.wikipedia.org/wiki/Dirichlet's_approximation_theorem Dirichlet's approximation theorem]
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| * [http://en.wikipedia.org/wiki/Probabilistic_method The Probabilistic Method]
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| * [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model Erdős–Rényi model for random graphs]
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| * [http://en.wikipedia.org/wiki/Graph_property Graph property]
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| * Some graph parameters: [http://en.wikipedia.org/wiki/Girth_(graph_theory) girth <math>g(G)</math>], [http://mathworld.wolfram.com/ChromaticNumber.html chromatic number <math>\chi(G)</math>], [http://mathworld.wolfram.com/IndependenceNumber.html Independence number <math>\alpha(G)</math>], [http://mathworld.wolfram.com/CliqueNumber.html clique number <math>\omega(G)</math>]
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| * [http://en.wikipedia.org/wiki/Extremal_graph_theory Extremal graph theory]
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| * [http://en.wikipedia.org/wiki/Turan_theorem Turán's theorem], [http://en.wikipedia.org/wiki/Tur%C3%A1n_graph Turán graph]
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| * Two analytic inequalities:
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| :*[http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality Cauchy–Schwarz inequality]
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| :* the [http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means inequality of arithmetic and geometric means]
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| * [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Stone_theorem Erdős–Stone theorem] (fundamental theorem of extremal graph theory)
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| * [http://en.wikipedia.org/wiki/Dirac's_theorem Dirac's theorem]
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| * [http://en.wikipedia.org/wiki/Hall's_theorem Hall's theorem ] (the marriage theorem)
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| * [http://en.wikipedia.org/wiki/Birkhoff-Von_Neumann_theorem Birkhoff-Von Neumann theorem]
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| * [http://en.wikipedia.org/wiki/K%C3%B6nig's_theorem_(graph_theory) König-Egerváry theorem]
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| * [http://en.wikipedia.org/wiki/Dilworth's_theorem Dilworth's theorem]
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| * [http://en.wikipedia.org/wiki/Sperner_family Sperner system]
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| * [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ko%E2%80%93Rado_theorem Erdős–Ko–Rado theorem]
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| * [http://en.wikipedia.org/wiki/VC_dimension VC dimension]
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| * [http://en.wikipedia.org/wiki/Kruskal%E2%80%93Katona_theorem Kruskal–Katona theorem]
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| * [http://en.wikipedia.org/wiki/Ramsey_theory Ramsey theory] | |
| :*[http://en.wikipedia.org/wiki/Ramsey's_theorem Ramsey's theorem]
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| :*[http://en.wikipedia.org/wiki/Happy_Ending_problem Happy Ending problem]
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| :*[http://en.wikipedia.org/wiki/Van_der_Waerden's_theorem Van der Waerden's theorem]
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| :*[http://en.wikipedia.org/wiki/Hales-Jewett_theorem Hales–Jewett theorem]
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| * [http://en.wikipedia.org/wiki/Lov%C3%A1sz_local_lemma Lovász local lemma]
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| * [http://en.wikipedia.org/wiki/Combinatorial_optimization Combinatorial optimization]
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| :* [http://en.wikipedia.org/wiki/Optimization_(mathematics) optimization]
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| :* [http://en.wikipedia.org/wiki/Convex_combination convex combination], [http://en.wikipedia.org/wiki/Convex_set convex set], [http://en.wikipedia.org/wiki/Convex_function convex function]
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| :* [http://en.wikipedia.org/wiki/Local_optimum local optimum] (see also [http://en.wikipedia.org/wiki/Maxima_and_minima maxima and minima])
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| * [http://en.wikipedia.org/wiki/Linear_programming Linear programming]
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| :* [http://en.wikipedia.org/wiki/Linear_inequality linear constraint]
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| :* [http://en.wikipedia.org/wiki/Hyperplane hyperplane], [http://en.wikipedia.org/wiki/Half_space halfspace], [http://en.wikipedia.org/wiki/Polyhedron polyhedron], [http://en.wikipedia.org/wiki/Convex_polytope convex polytope]
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| :* [http://en.wikipedia.org/wiki/Simplex_algorithm the Simplex algorithm]
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| * The [http://en.wikipedia.org/wiki/Max-flow_min-cut_theorem Max-Flow Min-Cut Theorem]
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| :* [http://en.wikipedia.org/wiki/Maximum_flow_problem Maximum flow]
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| :* [http://en.wikipedia.org/wiki/Minimum_cut minimum cut]
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| * [http://en.wikipedia.org/wiki/Unimodular_matrix Unimodularity]
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| * [http://en.wikipedia.org/wiki/Dual_linear_program Duality]
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| :* [http://en.wikipedia.org/wiki/Linear_programming#Duality LP Duality]
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| * [http://en.wikipedia.org/wiki/Matroid Matroid]
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| :* [http://en.wikipedia.org/wiki/Weighted_matroid weighted matroid] and [http://en.wikipedia.org/wiki/Greedy_algorithm greedy algorithm]
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| :* [http://en.wikipedia.org/wiki/Matroid_intersection Matroid intersection]
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| * [http://en.wikipedia.org/wiki/Laplacian_matrix Laplacian]
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| * [http://en.wikipedia.org/wiki/Algebraic_connectivity <math>\lambda_2</math> of a graph] and [http://en.wikipedia.org/wiki/Expander_graph#Cheeger_Inequalities Cheeger Inequalities]
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| * [http://en.wikipedia.org/wiki/Expander_graph Expander graph]
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| * [http://en.wikipedia.org/wiki/Szemer%C3%A9di_regularity_lemma Szemerédi regularity lemma]
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Template:Orphan
Graham's number is a very, very big natural number that was defined by a man named Ronald Graham. Graham was solving a problem in an area of mathematics called Ramsey theory. He proved that the answer to his problem was smaller than Graham's number.
Graham's number is one of the biggest numbers ever used in a mathematical proof. Even if every digit in Graham's number were written in the tiniest writing possible, it would still be too big to fit in the observable universe.
Context
Ramsey theory is an area of mathematics that asks questions like the following:
Template:Quote
It turns out that for this simple problem, the answer is "yes" when we have 6 or more points, no matter how the lines are colored. But when we have 5 points or fewer, we can color the lines so that the answer is "no".
Graham's number comes from a variation on this question.
Template:Quote
By asking that the 4 points lie on a plane, we have made the problem much harder. We would like to know: for what values of n is the answer "no" (for some way of coloring the lines), and for what values of n is it "yes" (for all ways of coloring the lines)? But this problem has not been completely solved yet.
In 1971, Ronald Graham and B. L. Rothschild found a partial answer to this problem. They showed that for n=6, the answer is "no". But when n is very large, as large as Graham's number or larger, the answer is "yes".
One of the reasons this partial answer is important is that it means that the answer is eventually "yes" for at least some large n. Before 1971, we didn't know even that much.
Definition
Graham's number is not only too big to write down all of its digits, it is too big even to write in scientific notation. In order to be able to write it down, we have to use Knuth's up-arrow notation.
We will write down a sequence of numbers that we will call g1, g2, g3, and so on. Each one will be used in an equation to find the next. g64 is Graham's number.
First, here are some examples of up-arrows:
- [math]\displaystyle{ 3\uparrow3 }[/math] is 3x3x3 which equals 27. An arrow between two numbers just means the first number multiplied by itself the second number of times.
- You can think of [math]\displaystyle{ 3 \uparrow \uparrow 3 }[/math] as [math]\displaystyle{ 3 \uparrow (3 \uparrow 3) }[/math] because two arrows between numbers A and B just means A written down a B number of times with an arrow in between each A. Because we know what single arrows are, [math]\displaystyle{ 3\uparrow(3\uparrow3) }[/math] is 3 multiplied by itself [math]\displaystyle{ 3\uparrow3 }[/math] times and we know [math]\displaystyle{ 3\uparrow3
}[/math] is twenty-seven. So [math]\displaystyle{ 3\uparrow\uparrow3 }[/math] is 3x3x3x3x....x3x3, in total 27 times. That equals 7625597484987.
- [math]\displaystyle{ 3 \uparrow \uparrow \uparrow 3 }[/math] is [math]\displaystyle{ 3 \uparrow \uparrow (3 \uparrow \uparrow 3) }[/math] and we know [math]\displaystyle{ 3\uparrow\uparrow3 }[/math] is 7625597484987. So [math]\displaystyle{ 3\uparrow\uparrow(3\uparrow\uparrow3) }[/math] is [math]\displaystyle{ 3\uparrow \uparrow 7625597484987 }[/math]. That can also be written as [math]\displaystyle{ 3\uparrow(3\uparrow(3\uparrow(3\uparrow . . .(3\uparrow(3\uparrow(3\uparrow3) }[/math] with a total of 7625597484987 3s. This number is so huge, its digits, even written very small, could fill up the observable universe and beyond.
- Although this number may already be beyond comprehension, this is barely the start of this giant number.
- The next step like this is [math]\displaystyle{ 3 \uparrow \uparrow \uparrow \uparrow 3 }[/math] or [math]\displaystyle{ 3 \uparrow \uparrow \uparrow (3 \uparrow \uparrow \uparrow 3) }[/math]. This is the number we will call g1.
After that, g2 is equal to [math]\displaystyle{ 3\uparrow \uparrow \uparrow \uparrow \ldots \uparrow \uparrow \uparrow \uparrow 3 }[/math]; the number of arrows in this number is g1.
g3 is equal to [math]\displaystyle{ 3\uparrow \uparrow \uparrow \uparrow \uparrow \ldots \uparrow \uparrow \uparrow \uparrow \uparrow 3 }[/math], where the number of arrows is g2.
We keep going in this way. We stop when we define g64 to be [math]\displaystyle{ 3\uparrow \uparrow \uparrow \uparrow \uparrow \ldots \uparrow \uparrow \uparrow \uparrow \uparrow 3 }[/math], where the number of arrows is g63.
This is Graham's number.
Related pages