We examine different methods of generalize Van der Waerden's Theorem, the Multidimensional Van der Waerden Theorem, the Canonical Van der Waerden Theorem, and other Variants. / Master of Science / Ramsey Theory is a subfield of mathematics in which randomness is studied from the perspective of partition regularity. We say that a structure <i>A</i> is partition regular within some space, if for any partition of the space into to some finite number of pieces, one of the pieces contains a copy of <i>A</i>. The simplest example of this, is letting <i>A</i> be the collection of 2 points sets, then no matter how you partition the integers into a finite number of pieces, at least one of the pieces must contain some 2 point set. If we replace 2 in the previous example with some fixed number <i>n</i>, then we obtain what is commonly referred to as the pigeon hole principle, which is one of the earliest results of combinatorics. To be more precise, the pigeon hole principle tells us that given any number <i>n</i>, and any finite partition of the positive integers, at least one of the pieces contains some <i>n</i> point set. However, the pigeon hole principle does not tell us anything about the <i>n</i> point set other than its size. Ramsey Theory seeks to generalize the pigeon hole principle by imposing further restrictions, by asking questions such as if we can always find an <i>n</i> point set consisting of consecutive integers, even integers, perfect squares, and so on. One of the resulting generalizations is known as Van der Waerden’s Theorem, which deals with structures known as arithmetic progressions. An arithmetic progression is a set of integers in which the difference between consecutive elements is the same, such as {3, 7, 11, 15, 19, 23, 27, 31}, or {a + jd}<sub>j=0</sub><sup>k</sup> is its most general form. Van der Waerden’s Theorem states that we can generalize the pigeon hole principle by assuming that the <i>n</i> point sets we are finding are also arithmetic progressions. To be more precise, Van der Waerden’s Theorem states that for any partition of the positive integers into a finite number of pieces, and any positive integer <i>n</i>, at least one of the pieces of the partition contains an arithmetic progression of <i>n</i> numbers. In this thesis, we will be examining how to further refine Van der Waerden’s Theorem and its generalizations.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/73355 |
| Date | 28 October 2016 |
| Creators | Farhangi, Sohail |
| Contributors | Mathematics, Brown, Ezra A., Mihalcea, Constantin Leonardo, Floyd, William J. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Thesis |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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