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13.3: Ramsey Theory - Mathematics


13.3: Ramsey Theory - Mathematics

Packages

This is the documentation for the Scala standard library.

This is the documentation for the Scala standard library.

Package structure

The scala package contains core types like Int , Float , Array or Option which are accessible in all Scala compilation units without explicit qualification or imports.

    and its sub-packages contain Scala's collections framework
      - Immutable, sequential data-structures such as Vector , List , Range , HashMap or HashSet - Mutable, sequential data-structures such as ArrayBuffer , StringBuilder , HashMap or HashSet - Mutable, concurrent data-structures such as TrieMap

    Other packages exist. See the complete list on the right.

    Additional parts of the standard library are shipped as separate libraries. These include:

      - Scala's reflection API (scala-reflect.jar) - XML parsing, manipulation, and serialization (scala-xml.jar) - Parallel collections (scala-parallel-collections.jar) - Parser combinators (scala-parser-combinators.jar) - A convenient wrapper around Java's GUI framework called Swing (scala-swing.jar)

    Automatic imports

    Identifiers in the scala package and the scala.Predef object are always in scope by default.

    Some of these identifiers are type aliases provided as shortcuts to commonly used classes. For example, List is an alias for scala.collection.immutable.List .

    Other aliases refer to classes provided by the underlying platform. For example, on the JVM, String is an alias for java.lang.String .

    Core Scala types. They are always available without an explicit import.

    This package object contains primitives for concurrent and parallel programming.

    This package object contains primitives for concurrent and parallel programming.

    Guide

    A more detailed guide to Futures and Promises, including discussion and examples can be found at http://docs.scala-lang.org/overviews/core/futures.html.

    Common Imports

    When working with Futures, you will often find that importing the whole concurrent package is convenient:

    When using things like Future s, it is often required to have an implicit ExecutionContext in scope. The general advice for these implicits are as follows.

    If the code in question is a class or method definition, and no ExecutionContext is available, request one from the caller by adding an implicit parameter list:

    This allows the caller of the method, or creator of the instance of the class, to decide which ExecutionContext should be used.

    For typical REPL usage and experimentation, importing the global ExecutionContext is often desired.

    Specifying Durations

    Operations often require a duration to be specified. A duration DSL is available to make defining these easier:

    Using Futures For Non-blocking Computation

    Basic use of futures is easy with the factory method on Future, which executes a provided function asynchronously, handing you back a future result of that function without blocking the current thread. In order to create the Future you will need either an implicit or explicit ExecutionContext to be provided:

    Avoid Blocking

    Although blocking is possible in order to await results (with a mandatory timeout duration):

    and although this is sometimes necessary to do, in particular for testing purposes, blocking in general is discouraged when working with Futures and concurrency in order to avoid potential deadlocks and improve performance. Instead, use callbacks or combinators to remain in the future domain:

    The package object scala.math contains methods for performing basic numeric operations such as elementary exponential, logarithmic, root and trigonometric functions.

    The package object scala.math contains methods for performing basic numeric operations such as elementary exponential, logarithmic, root and trigonometric functions.

    All methods forward to java.lang.Math unless otherwise noted.

    The package object scala.sys contains methods for reading and altering core aspects of the virtual machine as well as the world outside of it.

    The package object scala.sys contains methods for reading and altering core aspects of the virtual machine as well as the world outside of it.


    13.3: Ramsey Theory - Mathematics

    13. Distance Education
    PDF

    13.1 Introduction
    13.2 History of Distance Education
    13.3 Theory of Distance Education
    13.4 Distance Learning Technologies
    13.5 Future Directions and Emerging Technologies
    13.6 Research Related to Media in Distance Education
    13.7 International Issues
    13.8 Summary and Recommendations
    References

    13.3 THEORY OF DISTANCE EDUCATION

    The development of new technologies has promoted an astounding growth in distance education, both in the number of students enrolling and in the number of universities adding education at a distance to their curriculum (Garrison, 1990). While the application of modem technology may glamorize distance education, literature in the field reveals a conceptually fragmented framework lacking in both theoretical foundation and programmatic research. Without a strong base in research and theory, distance education has struggled for recognition by the traditional academic community. Distance education has been described by some (Garrison, 1990 Hayes, 1990) as no more than a hodgepodge of ideas and practices taken from traditional classroom settings and imposed on learners who just happen to be separated physically from an instructor. As distance education struggles to identify appropriate theoretical frameworks, implementation issues also become important. These issues involve the learner, the instructor, and the technology. Because of the very nature of distance education as learner-centered instruction, distance educators must move ahead to investigate how the learner, the instructor, and the technology collaborate to generate knowledge.

    Traditionally, both theoretical constructs and research studies in distance education have been considered in the context of an educational enterprise that was entirely separate from the standard, classroom-based, classical instructional model. In part to justify, and in part to explain, the phenomenon, theoreticians like Holmberg, Keegan, and Rumble explored the underlying assumptions of what it is that makes distance education different from traditional education. With an early vision of what it meant to be a nontraditional learner, these pioneers in distance education defined the distance learner as one who is physically separated from the teacher (Rumble, 1986), has a planned and guided learning experience (Holmberg, 1986), and participates in a two-way structured form of distance education that is distinct from the traditional form of classroom instruction (Keegan, 1988). In order to justify the importance of this nontraditional kind of education, early theoretical approaches attempted to define the important and unique attributes of distance education.

    Keegan (1986) identifies three historical approaches to the development of a theory of distance education. Theories of autonomy and independence from the 1960s and 1970s, argued by Wedemeyer (1977) and Moore (1973), reflect the essential component of the independence of the learner. Otto Peter's (1971) work on a theory of industrialization in the 1960s reflects the attempt to view the field of distance education as an industrialized form of teaching and learning. The third approach integrates theories of interaction and communication formulated by Badth (1982, 1987), and Daniel and Marquis (1979). Using the postindustrial model, Keegan presents these three approaches to the study and development of the academic discipline of distance education. It is this concept of industrialized, open, nontraditional learning that, Keegan says, will change the practice of education.

    Wedemeyer (1981) identifies essential elements of independent learning as greater student responsibility, widely available instruction, effective mix of media and methods, adaptation to individual differences, and a wide variety of start, stop, and learn times. Holmberg (1989) calls for foundations of theory construction around the concepts of independence, learning, and teaching:

    Meaningful learning, which anchors new learning matter in the cognitive structures, not rote learning, is the center of interest. Teaching is taken to mean facilitation of learning. Individualization of teaching and learning, encouragement of critical thinking, and far-reaching student autonomy are integrated with this view of learning and teaching (Holmberg, 1989, p. 161).

    Holmberg summarizes his theoretical approach by stating that :

    Distance education is a concept that covers the learning-teaching activities in the cognitive and/or psycho-motor and affective domains of an individual learner and a supporting organization. It is characterized by non-contiguous communication and can be carried out anywhere and at any time, which makes it attractive to adults with professional and social commitments (Holmberg, 1989, p. 168).

    Garrison and Shale (1987) include in their essential criteria for formulation of a distance education theory the elements of noncontiguous communication, two-way interactive communication, and the use of technology to mediate the necessary two-way communication.

    13.3.1 Theoretical Constructs

    Recently, a wider range of theoretical notions has provided a richer understanding of the learner at a distance. Four such concepts are transactional distance, interaction, learner control, and social presence.

    13.3.1.1. Transactional Distance. Moore's (1990) concept of "transactional distance" encompasses the distance that, he says, exists in all educational relationships. This distance is determined by the amount of dialogue that occurs between the learner and the instructor, and the amount of structure that exists in the design of the course. Greater transactional distance occurs when an educational program has more structure and less student-teacher dialogue, as might be found in some traditional distance education courses. Education offers a continuum of transactions from less distant, where there is greater interaction and less structure, to more distant, where there may be less interaction and more structure. This continuum blurs the distinctions between conventional and distance programs because of the variety of transactions that occur between teachers and learners in both settings. Thus distance is not determined by geography but by the relationship between dialogue and structure.

    Saba and Shearer (Saba & Shearer, 1994) carry the concept of transactional distance a step farther by proposing a system dynamics model to examine the relationship between dialogue and structure in transactional distance. In their study, Saba and Shearer conclude that as learner control and dialogue increase, transactional distance decreases. It is not location that determines the effect of instruction but the amount of transaction between learner and instructor. This concept has implications for traditional classrooms as well as distant ones. The use of integrated telecommunication systems may permit a greater variety of transactions to occur, thus improving dialogue to minimize transactional distance.

    13.3.1.2. Interaction. A second theoretical construct of recent interest to distance educators, and one that has received much attention in the theoretical literature, is that of interaction. Moore (1989) discusses three types of interaction essential in distance education. Learner-instructor interaction is that component of his model that provides motivation, feedback, and dialogue between the teacher and student. Learner-content interaction is the method by which students obtain intellectual information from the material. Learner-learner interaction is the exchange of information, ideas, and dialogue that occur between students about the course, whether this happens in a structured or nonstructured manner. The concept of interaction is fundamental to the effectiveness of distance education programs as well as traditional ones.Hillman, Hills, and Gunawardena (1994) have taken the idea of interaction a step farther and added a fourth component to the model learner-interface interaction. They note that the interaction between the learner and the technology that delivers instruction is a critical component of the model, which has been missing thus far in the literature. They propose a new paradigm that includes understanding the use of the interface in all transactions. Learners who do not have the basic skills required to use a communication medium spend inordinate amounts of time learning to interact with the technology and have less time to learn the lesson. For this reason, instructional designers must include learner-interface interactions that enable the learner to have successful interactions with the mediating technology.

    13.3.1.3. Control. A third theoretical concept receiving attention in the distance education literature is that of independence and learner control. Studies that examine locus of control (Altmann & Arambasich, 1982 Rotter, 1989) conclude that students who perceive that their academic success is a result of their own personal accomplishments have an internal locus of control and are more likely to persist in their education. Students with an external locus of control feel that their success, or lack of it, is due largely to events such as luck or fate outside their control. Thus, externals are more likely to become dropouts. Factors of control that influence dropout rate have been of concern to distance educators as they search for criteria to predict successful course completion. Baynton (1992) developed a model to examine the concept of control as it is defined by independence, competence, and support. She notes that control is more than independence. It requires striking a balance among three factors: a learner's independence (the opportunity to make choices), competence (ability and skill), and support (both human and material). Baynton's factor analysis confirms the significance of these three factors and suggests other factors that may affect the concept of control and which should be examined to portray accurately the complex interaction between teacher and learner in the distance learning setting.

    13.3.1.4. Social Context. Finally, the social context in which distance learning takes place is emerging as a significant area for research. Theorists are examining how the social environment affects motivation, attitudes, teaching, and learning. There is a widespread notion that technology is culturally neutral, and can be easily used in a variety of settings. However media, materials, and services are often inappropriately transferred without attention being paid to the social setting or to the local recipient culture (Mclsaac, 1993). Technology-based learning activities are frequently used without attention to -the impact on the local social environment. Computer-mediated communication attempts to reduce patterns of discrimination by providing equality of social interaction among participants who may be anonymous in terms of gender, race, and physical features. However, there is evidence that the social equality factor may not extend, for example, to participants who are not good writers but who must communicate primarily in a text-based format (Gunawardena, 1993). It is particularly important to examine social factors in distance learning environments where the communication process is mediated and where social climates are created that are very different from traditional settings. Feenberg and Bellman (1990) propose a social factor model to examine computer networking environments that create specialized electronic social environments for students and collaborators working in groups.

    One social factor particularly significant to distance educators is social presence, the degree to which a person feels "socially present" in a mediated situation. The notion is that social presence is inherent in the medium itself, and technologies offer participants varying degrees of "social presence' (Short, Williams & Christie, 1976). Hackman and Walker (1990), studying learners in an interactive television class, found that cues given to students such as encouraging gestures, smiles, and praise were social factors that enhanced both students' satisfaction and their perceptions of learning. Constructs such as social presence, immediacy, and intimacy are social factors that deserve further inquiry.

    13.3.2 Toward a Theoretical Foundation

    Although there have been numerous attempts to formulate a theory base for the field, American distance education remains "chaotic and confused. There is no national policy, nor anything approaching a consensus among educators of the value, the methodology or even the concept of distance education" (Moore, 1993). Shale (1990) calls for theoreticians and practitioners to stop emphasizing points of difference between distance and traditional education, but instead to identify common educational problems. Distance education is, after all, simply education at a distance with common frameworks, common conceptual concerns, and similar research questions relating to the social process of teaching and learning. Many distance educators are beginning to call for a theoretic model based on constructivist epistemology (Jegede, 1991). Technological advances have already begun to blur the distinction between traditional and distance educational settings. Time and place qualifiers are no longer unique. The need to test assumptions and hypotheses about how and under what conditions individuals learn best leads to research questions about learning, teaching, course design, and the role of technology in the educational process. As traditional, education integrates the use of interactive, multimedia technologies to enhance individual learning, the role of the teacher changes from knowledge source to knowledge facilitator. As networks become available in schools and homes to encourage individuals to become their own knowledge navigators, the structure of education will change, and the need for separate theories for distance education will blend into the theoretical foundations for the mainstream of education.

    More than 35% of the literature reviewed reported the need for developing a central, theoretical framework on which future distance education development can be based. While numerous journal articles and conference presentations discussed the lack of theoretical framework in the field, most of the work was descriptive rather than research oriented. However, several writers have contributed to theory formulation.

    Verduin and Clark (1991) offer a rationale by suggesting that confusion over distance education terminology may be to blame. In response to this theoretical void, Gibson (1990) suggests borrowing a theory from existing disciplines. Miller (1989) concurs by suggesting that "it is important that the study of distance education be informed by work done in other disciplines" (p. 15). Boyd and Apps (1980) struggle with the idea of borrowing a theory, as they see the important issue being the development of a clearly defined structure, function, purpose, and goal for distance education. "We must ask ourselves what erroneous assumptions we may be accepting when we borrow from established disciplines to define distance education" (pp. 2-3). Furthermore, borrowing extensively from other fields in order to define and solve problems allows the field to define the borrowed field (Gibson, 1990) In an effort to define theoretically the field of distance education, the literature advances three strategies. Deshler and Hagen (1989) advocate a multidisciplinary and interdisciplinary approach resulting in a diversity of perspectives. They caution that anything short of this approach may "produce theory that suffers from a view that is narrow, incomplete, discipline-based and restricted . to a predominant view of reality" (p. 163).

    A second approach is advocated by Hayes (1990), who supports the work of Knowles (1984) and Brookfield (1986). Hayes emphasizes that theoretical development relative to adult learning must be distinct from youth learning. While past experiences may occasionally interfere with an adult's openness to new learning experiences, the majority of literature views experience as a resource for new learning. Knowles (1984), for example, supports an andragogical, learner-focused foundation in his belief that "adults draw on previous experiences in order to test the validity of new information" (p. 44). A third strategy for theory development from an international perspective has been proposed by Sophason and Prescott (1988). They caution that certain lines of questioning are more appropriate in some countries than in others, thus the emanating theory "may have a particular slant" (p. 17). A comparative analysis strategy would undoubtedly be influenced by cultural bias and language barriers (Pratt, 1989). Pratt further indicates that understanding different culturally related beliefs about the nature of the individual and society may be critical in defining appropriate distance education theories. Pratt clarifies his belief through a description of how differences in societies' historical traditions and philosophies can contribute to differing orientations toward self-expression and social interactions within educational settings.

    Although these three strategies for the advancement of a theoretical foundation for distance education are repeated in current literature, Ely (1992) foresees a road block to the theoretical progression. "What seems to be needed is an unclouded understanding of distance education. This includes the audience, setting, and delivery methodologies" (p. 43). Loesch and Foley (1988) concur and ask for further research in this area in their statement that only when a clear understanding of distance education becomes available can concise questions be developed that can lead to establishment of theory. Evans and Nation (1992) contribute some of the most thoughtful and insightful comments on theory building when they suggest that we examine broader social and historic contexts in our efforts to extend previously narrow views of theories in open and distance education. They urge us to move toward deconstruction of the instructional industrialism of distance education, and toward the construction of a critical approach that, combined with an integration of theories from the humanities and social sciences, can enrich the theory building in our field.

    Although there has been no central theoretical framework to guide research in distance education, there have been a number of important studies that have examined the interactions of technologies with learning, course design, ,and instruction. Because of the heavy use of technology in distance education, it is appropriate to examine its role in this context.


    Updated August 3, 2001
    Copyright © 2001
    The Association for Educational Communications and Technology

    AECT
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    The anti-woo stuff [ edit ]

    To witness Ramsey theory in full effect, a case study is illustrative.

    In 2010 that fine bastion of objective British journalism, the Daily Mail [note 1] , published an article entitled "How a prehistoric sat nav stopped our ancestors getting lost in Britain" Ώ] . The article presented research which analysed the location of 1,500 prehistoric monuments and found them all to be on a grid of isosceles triangles, with each pointing to the next. According to the researcher Tom Brooks, "Such patterns could only have been the work of highly intelligent surveyors and planners which throws into question all previous claims as to the origin of mathematics."

    There is no doubt that Brooks did actually find the "mysterious" grid lines. Unfortunately, he did so by skipping over the vast majority of the sites, choosing only the few that happened to line up. Matt Parker of the University of London's School of Mathematics subsequently determined that there were 561,375,500 different possible "ley line triangles" using the 1500 monuments as a basis and that there was nowhere you could stand in the entire British Isles that was more than 58 metres away from a ley line intersection.

    Parker later went on to "prove" the same theory about a prehistoric navigation system using the location of Woolworths supermarkets ΐ] . He justified his research on the basis that, "if we analysed the sites we could learn more about what life was like in 2008 and how these people went about buying cheap kitchen accessories and discount CDs".

    Ramsey theory says that such "apparent order" is not only likely, but that as the number of member elements increases this "apparent order" actually becomes unavoidable. It is important to note that of all the total permutations, those which appear ordered will only represent a minuscule proportion. Hence a key aspect of woo arguments that exploit Ramsey theory is the fact that the vast majority of data is ignored in favour of the tiny set which meets whatever "apparent order" was wanted.


    Undergraduate Math Student Pushes Frontier of Graph Theory

    Ashwin Sah stands in AmberGlen park, near his hometown of Portland, Oregon, where he’s been spending time during the pandemic.

    Kevin Hartnett

    On May 19, Ashwin Sah posted the best result ever on one of the most important questions in combinatorics. It was a moment that might have called for a celebratory drink, only Sah wasn’t old enough to order one.

    The proof joined a long list of mathematical results that Sah, who turned 21 in November, published while an undergraduate at the Massachusetts Institute of Technology (he posted this new proof just after graduating). It’s a rare display of precocity even in a field that celebrates youthful genius.

    “He has done enough work as an undergraduate to get a faculty position,” said David Conlon of the California Institute of Technology.

    The May proof focused on an important feature of combinatorics called Ramsey numbers, which quantify how big a graph (a collection of dots, or vertices, connected by edges) can get before it necessarily contains a certain kind of substructure.

    For example, imagine you’ve got six vertices, each connected to every other vertex by edges. Now color each of the 15 total edges either red or blue. No matter how you apply the colors, it’s inevitable that you’ll end up with three vertices that are all connected to each other by edges of the same color (known as a “clique”). The same is not true, however, if you start with five vertices (for which it’s possible to do the coloring without creating a clique). As a result, mathematicians say that the Ramsey number for two colors and a clique of size 3 is 6 — meaning you need at least six vertices to guarantee the clique exists.

    As the size of the clique you’re looking for grows bigger, it becomes very difficult to calculate exact Ramsey numbers. Instead, mathematicians try to zero in on them by guaranteeing that the Ramsey number for a clique of some arbitrary size is greater than some number (the “lower bound”) and less than another (the “upper bound”).

    Paul Erdős and George Szekeres initiated the study of upper and lower bounds for Ramsey numbers in the 1930s. Since then, mathematicians have made relatively little progress on either one — though Quanta recently covered an innovative new proof that set the best-ever lower bound for some Ramsey numbers.


    A More Mathematical Explanation

    The answer we just found is called a Ramsey number. In the simplistic terms of the party proble [. ]

    The answer we just found is called a Ramsey number. In the simplistic terms of the party problem, a Ramsey number R(m,n) is the minimum number of people you must invite so that at least m people will be mutual friends or at least n people will be mutual strangers. In the previous case, we proved that 6 is the Ramsey number when you want at least 3 people to be either mutual acquaintances or mutual strangers. In other words, R(3,3) = 6.

    Since R(3,3) = 6, we can be specific and say that v ≥ 6.

    As we will see later on, there is a Ramsey number for all complete graphs—there exists an R(3,4), R(4,4), etc.


    Math 497A - Introduction to Ramsey Theory

    Course blog: Links to supplementary material, hints to homework problems etc will be posted on http://massramsey2011.wordpress.com.

    Content

    The course will cover some central results of Ramsey Theory. The basic paradigm of Ramsey theory is that if a structure is sufficiently large, it will have very regular substructures of a certain size. We will illustrate this principle by means of a number of results from graph theory, number theory, and combinatorial geometry. Along the way, we will encounter a phenomenon typical of Ramsey theory -- sufficiently large often means really large. We will investigate this phenomenon and see that it has some interesting consequences concerning the foundations of mathematics.

    Lecture Notes

    Recommended Reading

    • Graham, Rothschild, and Spencer – Ramsey Theory, 1990
    • Nesetril – Ramsey Theory, in: Handbook of Combinatorics, 1995

    Homework

    Homework will be assigned each Monday and will be due in class the following Monday in class. Homework will be graded and the two lowest scores will be dropped. Late homework will not be accepted. There will be no exception to this rule. Of course it may happen that you cannot turn in homework because you were ill or for some other valid reason. This is why the two lowest scores will be dropped.


    Homework 1, due August 29, 2011 (Solutions)
    Homework 2, due September 7, 2011 (Solutions)
    Homework 3, due September 12, 2011 (Solutions)
    Homework 4, due September 19, 2011 (Solutions)
    Homework 5, due September 26, 2011 (Solutions)
    Homework 6, due October 3, 2011 (Solutions)
    Homework 7, due October 24, 2011
    Homework 8, due October 31, 2011
    Homework 9, due November 7, 2011
    Homework 10, due November 14, 2011 (Solutions)
    Homework 11, due December 5, 2011

    Research Project

    Each participant is required to complete a research project on a specific topic. This will usually include reading original research papers. As part of the final exam, each participant will give a 20 minute representation on his/her project. Furthermore, participants are required to prepare a 5-10 page written report.

    I will make available a list of possible projects in October. However, I welcome suggestions from students. So, look around, read a bit, maybe you will find a topic that interests you particularly.

    Exams

    There will be a midterm: Monday, Oct 10, 10:10-12.
    Midterm preparation sheet (Oct 5, 2011)
    This exam will be a closed book exams. No cheat sheets! Bring blue books.


    The final exam will, according to MASS tradition, be an individual 1 hour oral exam for each participant.

    Grading Policy

    The final grade will take into account the homework scores, the midterm, the research project, and the final oral exam.

    Academic Integrity

    Collaboration: Collaboration among students to solve homework assignments is welcome. This is a good way to learn mathematics. So is the consultation of other sources such as other textbooks. However, every student has to hand in his/her own set of solutions, and if you use other people's work or ideas you have to indicate the source in your solutions.
    (In any case, complete and correct homework receives full credit.)

    However, from time to time there will be "controlled" problems, in which every student should work out his/her own solutions.


    On-Line Ramsey Theory (2004)

    Definitions: Builder and Painter play a game on graphs. In each round, Builder adds an edge (infinitely many vertices are available), and Painter colors it red or blue. Builder wins by forcing Painter to produce a monochromatic copy of a fixed graph G. Painter wins by forever avoiding that.

    Background: The game is closely related to graph Ramsey theory. Ramsey's Theorem guarantees that when n is sufficiently large, every red/blue coloring of the complete graph Kn contains a monochromatic copy of G). The least such n is the Ramsey number R(G).

    Without restrictions on the graph Builder presents, Builder thus wins by presenting a sufficiently large complete graph. Hence the rules of the game also specify a graph family H such that after every move the graph that has been built must lie in H. We specify a particular on-line Ramsey game as (G,H). is unavoidable on F if Builder wins (F,G) otherwise G is avoidable on F. A family F is self- unavoidable if Builder wins (F,G) for every G &isin F. !-->

    1. Builder wins (G,H) when G is a forest and H is the family of all forests. (This has an easy proof by induction.)
    2. Builder wins (G,H) when G is k-colorable and H is the family of all k-colorable graphs. (This is much more difficult and uses the bipartite version of Ramsey's Theorem.)
    3. Painter wins (C3,H) when H is the family of outerplanar graphs.
    4. Builder wins (C3,H) when H is the family of 2-degenerate planar graphs.
    5. Builder wins (G,H) when H is the family of planar graphs and G consists of a single cycle plus any set of chords incident to a single vertex.

    Conjecture 1: When H is the family of planar graphs, Builder wins (G,H) if and only if G is outerplanar. (It is unknown whether Builder can win on this family for any graph G that is not outerplanar, and it is not known whether Painter can win it for any G that is outerplanar.)

    Problem 2: Given a monotone graph parameter &rho, let Hk= . Let f(k) be the minimum m such that Builder wins (G,Hm) whenever G&isinHk. For which graph parameters is it true that f(k) is finite for all k? Determine f(k) for some values of k for some parameter &rho. In particular, is f a bounded function when &rho is "degeneracy".

    Comments: Monotonicity for &rho is the property that Hk&subHk+1 for all k. Theorem 2 from [GHK] shows that f(k)=k when the parameter is "chromatic number". Theorem 1 from [GHK] shows that f(1)=1 when the parameter is "degeneracy". In general, can Builder force any k-degenerate graph when playing k-degenerate graphs? This scenario also has been studied for &rho= "number of edges" in [GKP] and for &rho= in the 2007 REGS group (see below).

    Definition: We use osr(G) to denote the on-line (size) Ramsey number of a graph G, defined to be the minimum m such that Builder can force G by playing in the class of graphs with at most m edges. Simply put, osr(G) is the number of edges Builder may need to play to force Painter to make a monochromatic copy of G when there is no restriction on the edges to be played (other than how many there are.)

    Comments: In ordinary Ramsey theory, the size Ramsey number of a graph G is the minimum number of edges in a graph H such that every 2-edge-coloring of H contains a monochromatic copy of G. Clearly osr(G) (defined in [GKP] without putting "size" in the term) is bounded by the size Ramsey number of G. Beck [B] gave a relatively easy proof that osr(Kpp)&ge½R(Kp).

    Question 3: Is osr(Kp) subquadratic in R(Kp)?

    Question 4: Does osr(G) &ge R(G)/2 hold for all G? For what graphs is osr(G) linear in R(G)? For which graphs can osr(G) be computed?

    Comments: Question 3 may be hard, while progress on parts of Question 4 may be easy. Note that osr(K1,m)=2m-1. [GKP] studied this parameter for paths, trees, and some other graphs. They proved for example that osr(Pn)&le 4n-7, with exact values up to P6, but also that osr(G) can be quadratic in the number of vertices when G is a tree.

    Definition: We use odr(G) to denote the on-line degree Ramsey number of a graph G, defined to be the minimum k such that Builder can force G by playing in the class Sk of graphs with maximum degree at most m.

    1. odr(G)&le3 if and only if each component of G is a path or each component of G is a subgraph of K1,3.
    2. odr(G)&le2&Delta(G)-1 when G is a tree.
    3. odr(G)&ge&Delta(G)-1+maxuv&isinE(G)min
    4. odr(Cn)=4 if n is even or at least 689 or equal to 3, and it is always at most 5.
    5. odr(G)&le8 if &Delta(G)&le2

    Question 6: What is odr(C5)? Does odr(Cn) equal 4 for all n?

    Question 7: Can odr(G) be bounded in terms of &Delta(G)? (This is a special case of Problem 3.)

    Comments: One could also consider games that are easier for Builder by enlarging the target H also to a family Builder wins by forcing a monochromatic copy of any graph in H. For example, on what families is <>3,C4> unavoidable? Other related problems were studied in [B] and [FKRRT].

    References:
    [B] Beck, J. Achievement games and the probabilistic method. Combinatorics, Paul Erd?? is eighty, Vol. 1, 51--78, Bolyai Soc. Math. Stud., J??os Bolyai Math. Soc., Budapest, 1993.
    [FKRRT] Friedgut, E. Kohayakawa, Y. R??l, V. Ruciński, A. Tetali, P. Ramsey games against a one-armed bandit. Special issue on Ramsey theory. Combin. Probab. Comput. 12 (2003), no. 5-6, 515--545.
    [GHK] Grytczuk, J. A. Hałuszczak, M. Kierstead, H. A. On-line Ramsey theory. Electron. J. Combin. 11 (2004), no. 1, Research Paper 57, 10 pp. Int. J. Math. Comput. Sci. 1 (2006), no. 1, 117--124.


    Chapter 33: Ramsey Theory

    Ramsey Theory in its entirety is too complicated to be explained within The Colossal Book of Mathematics mentioned in the book. However, Ramsey Theory can be applied much easier to a recreational view on graph-coloring theory games. The most famous Ramsey game is known as Sim, which was named after the mathematician Gustavus Simmons. Sim is directly related to the problem, “Prove that at a gathering of any six people, some three of them are either mutual acquaintances or complete strangers to each other.” This would be accomplished by being played on a complete graph with six separate points known as (K6), and having two different colors (ex. Blue & Red) to represent a mutual acquaintance connection (Blue) or complete strangers to each other (Red) between the six individuals. Players would then take alternate turns connecting a point with their respective color. The goal is to avoid completing a monochromatic triangle of your color (either blue or red). Therefore, the loser is the first person to connect 3 separate points of his or her own color. In Ramsey Theory talk, the purpose of this is to prove that it is impossible not to complete either sub graph of an entirely red triangle (K3) or entirely blue triangle (K3). Meaning, eventually you will complete an entirely red triangle or entirely blue triangle. In Ramsey Theory referring to actual numbers and graphs, the above problem can be expressed as R(3,3) = 6. In the notation, R represents Ramsey number, the first 3 for one of the colored triangles, the second 3 for the other colored triangle, and the 6 for representing the smallest number of points, which both a red or blue triangle is forced.


    These ideas of classical Ramsey numbers and theory can lead us to different games. There is the avoidance game or achievement game (known as Sim), and then there is an alternate game which leads to who can complete a larger amount of K3 monochromatic triangles or who can complete the least amount. Apart from the game R(3,3) = 6, other much more challenging Ramsey games can be played. When changing the values in R(r,s)= K n , games get much more challenging. If the K 6 value is increased (ex. R(3,3) = 7) then there are now 7 points of which you play the game. If you were to increases K 3 value (ex. R(4,4) = 18) then it also increases the minimum amount to K 18 points, and changes the forced complete sub graph to a K 4 monochromatic tetrahedron. However, the jump from R(4,4) to R(5,5) is too complex to determine a K n complete graph for R(5,5). Stefan Burr, a leading expert on Ramsey theory estimates that the K n value will never be determined because it is too difficult to analyze.

    1 comment:

    I found this chapter to be rather difficult to understand, but you did a good job explaining the main point of the Ramsey Game and the how classical Ramsey numbers and theory can lead to different games. I think it would be fun to give Ramsey’s (K6) game a try, using the same two color method with six people, three mutual acquaintances and three complete strangers. Where it started to get confusing for me was at the part when the values of R(r,s)=Kn are increased and the graphs become more complex with more variations of Paths, Cycles, Starts and Wheels appearing.
    The part in Chapter 33 on Ramsey Theory that really interested me was the part on the General case of wheels. “The Ramsey number for the wheel of four points, the tetrahedron, is, as we have seen 18. The wheel of five points was shown to have a Ramsey number of 15 by Tim Moon, a Nigerian mathematician.” The six-point wheel’s Ramsey number in this sequence has yet to be solved, mathematicians estimate its number being between 17 and 20. The fact that this has theory has been around since 1950’s and we are still working on solving some of the wheel functions is fascinating to me.


    Ramsey Theory, 2nd Edition

    Covering all the major concepts, proofs, and theorems, theSecond Edition of Ramsey Theory is the ultimate guideto understanding every aspect of Shelah's proof, as well asthe original proof of van der Waerden. The book offers a historicalperspective of Ramsey's fundamental paper from 1930 andErdos' and Szekeres' article from 1935, while placingthe various theorems in the context of T. S. Motzkin'sthought on the subject of "Complete Disorder isImpossible."

    Ramsey Theory, Second Edition includes new and excitingcoverage of Graph Ramsey Theory and Euclidean Ramsey Theory andalso relates Ramsey Theory to other areas in discrete mathematics.In addition, the book features the unprovability results of Parisand Harrington and the methods from topological dynamics pioneeredby Furstenburg.

    Featuring worked proofs and outside applications, RamseyTheory, Second Edition addresses:
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    * Regular homogeneous and nonhomogeneous systems andequations
    * Special cases and broader interdisciplinary applications ofRamsey Theory principles

    An invaluable reference for professional mathematicians workingin discrete mathematics, combinatorics, and algorithms, RamseyTheory, Second Edition is the definitive work on thesubject.


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