On Sets of Acquaintances and Strangers at any Party

Goodman, A. W.

doi:10.1080/00029890.1959.11989408

Cited by 207 publications

(105 citation statements)

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“…Ref. [11] derived the minimum number of 3-cliques and 3-independent sets that can be present in an N -vertex graph. This minimum is precisely our E gs for R(3, 3) and a given N .…”

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confidence: 99%

Ramsey Numbers and Adiabatic Quantum Computing

Gaitan

Clark

2012

Phys. Rev. Lett.

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The graph-theoretic Ramsey numbers are notoriously difficult to calculate. In fact, for the twocolor Ramsey numbers R(m, n) with m, n ≥ 3, only nine are currently known. We present a quantum algorithm for the computation of the Ramsey numbers R(m, n). We show how the computation of R(m, n) can be mapped to a combinatorial optimization problem whose solution can be found using adiabatic quantum evolution. We numerically simulate this adiabatic quantum algorithm and show that it correctly determines the Ramsey numbers R(3, 3) and R(2, s) for 5 ≤ s ≤ 7. We then discuss the algorithm's experimental implementation, and close by showing that Ramsey number computation belongs to the quantum complexity class QMA.PACS numbers: 03.67. Ac,02.10.Ox,89.75.Hc In an arbitrary party of N people one might ask whether there is a group of m people who are all mutually acquainted, or a group of n people who are all mutual strangers. Using Ramsey theory [1,2], it can be shown that a threshold value R(m, n) exists for the party size N so that when N ≥ R(m, n), all parties of N people will either contain m mutual acquaintances, or n mutual strangers. The threshold value R(m, n) is an example of a two-color Ramsey number. Other types of Ramsey numbers exist, though we will focus on two color Ramsey numbers in this paper.One can represent the N -person party problem by an N -vertex graph. Here each person is associated with a vertex, and an edge is drawn between a pair of vertices only when the corresponding people know each other. In the case where m people are mutual acquaintances, there will be an edge connecting any pair of the m corresponding vertices. Similarly, if n people are mutual strangers, there will be no edge between any of the n corresponding vertices. In the language of graph theory [3], the m vertices form an m-clique, and the n vertices form an nindependent set. The party problem is now a statement in graph theory: if N ≥ R(m, n), every graph with N vertices will contain either an m-clique, or an n-independent set. Ramsey numbers can also be introduced using colorings of complete graphs, and R(m, n) corresponds to the case where only two colors are used.Ramsey theory has found applications in mathematics, information theory, and theoretical computer science [6]. An application of fundamental significance appears in the Paris-Harrington (PH) theorem of mathematical logic [4] which established that a particular statement in Ramsey theory related to graph colorings and natural numbers is true, though unprovable within the axioms of Peano arithmetic. Such statements are known to exist as a consequence of Godel's incompleteness theorem, though the PH theorem provided the first natural example. Deep connections have also been shown to exist between Ramsey theory, topological dynamics, and ergodic theory [5].Ramsey numbers grow extremely quickly and so are notoriously difficult to calculate. In fact, for two color Ramsey numbers R(m, n) with m, n ≥ 3, only nine are presently known [3]. To check whether N ? = R(m, n) requir...

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“…Ref. [11] derived the minimum number of 3-cliques and 3-independent sets that can be present in an N -vertex graph. This minimum is precisely our E gs for R(3, 3) and a given N .…”

mentioning

confidence: 99%

Ramsey Numbers and Adiabatic Quantum Computing

Gaitan

Clark

2012

Phys. Rev. Lett.

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“…In this section we consider only simple ordinary graphs, and we are interested in the functions g The first general bound on g r (x) was proved by Goodman [Goo59]:…”

Section: Clique Densitiesmentioning

confidence: 99%

Flag Algebras: An Interim Report

Razborov

2013

The Mathematics of Paul Erdős II

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For the most part, this article is a survey of concrete results in extremal combinatorics obtained with the method of flag algebras. But our survey is also preceded, interleaved and concluded with a few general digressions about the method itself. Also, instead of giving a plain and unannotated list of results, we try to divide our account into several connected stories that often include historical background, motivations and results obtained with the help of methods other than flag algebras. A forewordWhen I was asked by the organizers to contribute something on flag algebras, I was a bit uncertain at first. The reasons will become clear from the text below, but a two-sentence summary is this. In just a few recent years we have witnessed a tremendous explosion of activity in this area, and the explosion is still ongoing. It does not look (at least to me) quite consistent with the inevitable stamp of finality a full-fledged survey is supposed to convey.As a consequence, this contribution has a very clear flavor of an accounting book. I will try my best to summarize in Section 3, in a categorized and

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“…It turns out there are always at least two monochromatic triangles in K 6 [3,4]! Since there is at least one monochromatic triangle in K 6 , assume it is formed by the edges H1, 2L, H1, 3L, and H2, 3L and, by symmetry, assume that its edges are all red.…”

Section: Using Boolean Computation To Solve Some Problems From Ramseymentioning

confidence: 99%

Using Boolean Computation to Solve Some Problems from Ramsey Theory

Cowen¹

2013

TMJ

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Using some examples from Ramsey theory, this article shows how to use Mathematicaʼs Boolean computational capability. ‡ Introduction Mathematica's industrial-strength Boolean computation capability is not used as often as it should be. There probably are several reasons for this lack of use, but it is our view that a primary reason is lack of experience in expressing mathematical problems in the form required for Boolean computation. We look at a typical problem that is susceptible to Boolean analysis and show how to translate it so that it can be tested for satisfiability with Mathematica's built-in function SatisfiableQ. The problems we investigate come from an area of mathematics called Ramsey theory. Although Ramsey theory has been studied extensively for over 80 years and still provides many challenges, we neglect the theory (for the most part) and instead concentrate on translating the problems so that they are amenable to Boolean computation and then see what can be accomplished by computation alone. Those interested in learning a little more about Ramsey theory can consult [1]; for a standard reference, see [2]. ‡ Boolean RepresentationWe only concern ourselves with Boolean formulas in conjunctive normal form (cnf). A cnf is a conjunction of disjunctions of statements or their negations; the disjuncts are often called clauses.An example of a cnf is HA Í Ÿ B Í CL Ï HŸ A Í B Í CL Ï HŸ A Í Ÿ B Í Ÿ CL; letters represent statements and the symbols fi, fl, and Ÿ stand for "or," "and," and "not." A propositional formula is satisfiable if there is an assignment of true and false to its statements that makes the formula true when evaluated using the usual truth table rules.The Mathematica Journal 15

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On Sets of Acquaintances and Strangers at any Party

Cited by 207 publications

References 3 publications

Ramsey Numbers and Adiabatic Quantum Computing

Ramsey Numbers and Adiabatic Quantum Computing

Flag Algebras: An Interim Report

Using Boolean Computation to Solve Some Problems from Ramsey Theory

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