## Publication: On aspects of Ramsey theory

 dc.contributor.advisor Britz, Thomas en_US dc.contributor.advisor Greenhill, Catherine Suzanne en_US dc.contributor.author Chng, Zhi Yee en_US dc.date.accessioned 2022-03-22T17:41:57Z dc.date.available 2022-03-22T17:41:57Z dc.date.issued 2018 en_US dc.description.abstract This thesis presents various types of results from Ramsey Theory, most particularly, Ramsey-type theorems concerning graphs and families of sets. This thesis consists of 8 chapters. In Chapter 1, we give a brief historical introduction to Ramsey Theory. Then, we introduce some necessary notation and definitions that will be consistently used throughout the thesis, including some basic knowledge of Graph Theory which is particularly useful in Chapters 2 and 3. We present Ramsey-type results about graphs in Chapters 2 and 3. In Chapter 2, we introduce the classical Ramsey's Theorem which is the Ramsey-type theorem on the edge-colouring of the complete graph. We also introduce Ramsey numbers and present some results on these, especially some upper and lower bounds. In Chapter 3, we look at Ramsey-type results for monochromatic tree graphs, cycle graphs and bipartite graphs, respectively, occurring in arbitrary edge colourings of the complete graph. Then, we present the bipartite version of Ramsey's Theorem. Chapters 4, 5 and 6 present other famous Ramsey-type theorems, for arithmetic progressions and other, more general, structures. In Chapter 4, we introduce and prove Van der Waerden's Theorem and we also present some results on the bounds of the Van der Waerden numbers. In Chapter 5, we present Schur's Theorem and some results relating to the Schur numbers. Then, we look into some generalisations of Schur's Theorem, including Rado's Theorem and Folkman's Theorem. In Chapter 6, we prove the Hales-Jewett Theorem. We also construct a proof of Van der Waerden's Theorem by using the Hales-Jewett Theorem. Before we end our studies, in Chapter 7, we include some application of the Ramsey Theory. We look into the application of the Ramsey Theory in various fields, including graph theory, geometry and number theory. In Chapter 8, we conclude our studies. We give some overall comment on Ramsey Theory and include some possible future work on the field. en_US dc.identifier.uri http://hdl.handle.net/1959.4/60220 dc.language English dc.language.iso EN en_US dc.publisher UNSW, Sydney en_US dc.rights CC BY-NC-ND 3.0 en_US dc.rights.uri https://creativecommons.org/licenses/by-nc-nd/3.0/au/ en_US dc.subject.other Van der Waerden's theorem en_US dc.subject.other Ramsey theory en_US dc.subject.other Ramsey theorem en_US dc.subject.other Schur's theorem en_US dc.title On aspects of Ramsey theory en_US dc.type Thesis en_US dcterms.accessRights open access dcterms.rightsHolder Chng, Zhi Yee dspace.entity.type Publication en_US unsw.accessRights.uri https://purl.org/coar/access_right/c_abf2 unsw.identifier.doi https://doi.org/10.26190/unsworks/20521 unsw.relation.faculty Science unsw.relation.originalPublicationAffiliation Chng, Zhi Yee, Mathematics & Statistics, Faculty of Science, UNSW en_US unsw.relation.originalPublicationAffiliation Britz, Thomas, Mathematics & Statistics, Faculty of Science, UNSW en_US unsw.relation.originalPublicationAffiliation Greenhill, Catherine Suzanne, Mathematics & Statistics, Faculty of Science, UNSW en_US unsw.relation.school School of Mathematics & Statistics * unsw.thesis.degreetype Masters Thesis en_US
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