Asfaw, Amsalu (2014) Ramsey Theory. Masters thesis, Addis Ababa University.
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PDF (Ramsey Theory)
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Abstract
Ramsey Theory is named after Frank Plumpton Ramsey a young man was especially interested in logical and philosophy. Ramsey died at the age of 26 in 1930 the same year that his paper on a problem of formal logic was published. His paper catalyzed the development of the mathematics field now known as Ramsey Theory. Problem in Ramsey Theory typically ask a question of the form: How many people are required at gathering so that there must exist either three mutual acquaintances or three mutual strangers? We can rephrase this question as a problem in Ramsey Theory: How many vertices do you need in edge 2 colored complete graphs for it to be necessary that there be either a red K3 (people who know each other) or a blue K3 (people who do not know each other)? While many results in the subjects are published each year, there are many questions whose answer remains elusive. Ramsey Theory has played an important role in a plethora of mathematics development throughout the last century. Ramsey Theory is concerned with the preservation of structure under partition it is the study of unavoidable regularity in large structures. In this project, I explore some of the core ideas underpinning Ramsey Theory and present a variety of problem to which it can provide interesting and elegant solution. Also we have see, the Ramsey number R(k; l) is the smallest natural number n such that in any red and blue coloring of the edges of the complete graph on n vertices, we are guaranteed to find either a red Kk or a blue Kl. Furthermore, we have discussed a multi-color example R(3; 3; 3) = 17, generalization of Ramsey Theorem and infinite Ramsey Theory. Some Known Ramsey Theorem for bound on classical Ramsey numbers are included.
| Item Type: | Thesis (Masters) |
|---|---|
| Subjects: | Q Science > Q Science (General) Q Science > QA Mathematics |
| Divisions: | Africana |
| Depositing User: | Selom Ghislain |
| Date Deposited: | 15 Aug 2018 13:09 |
| Last Modified: | 15 Aug 2018 13:09 |
| URI: | http://thesisbank.jhia.ac.ke/id/eprint/4755 |
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