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Ramsey Theory and 2-Colorings of 3-Dimensional Grids.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Ramsey Theory and 2-Colorings of 3-Dimensional Grids./
Author:	Miller, Alexander.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2022,
Description:	38 p.
Notes:	Source: Masters Abstracts International, Volume: 83-12.
Contained By:	Masters Abstracts International83-12.
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=29208056
ISBN:	9798819365564

Ramsey Theory and 2-Colorings of 3-Dimensional Grids.
Miller, Alexander.

Ramsey Theory and 2-Colorings of 3-Dimensional Grids. - Ann Arbor : ProQuest Dissertations & Theses, 2022 - 38 p.

Source: Masters Abstracts International, Volume: 83-12.

Thesis (M.S.)--Lamar University - Beaumont, 2022.

This item must not be sold to any third party vendors.

In 1928 the English mathematician Frank Plumpton Ramsey proved that patterns are actually implicit in any large structure. In fact, Ramsey theory states that any structure will necessarily contain an orderly substructure. As the late American mathematician Theodore S. Motzkin first proclaimed, Ramsey theory implies that complete disorder is an impossibility. In their paper "Rectangle Free Coloring of Grids,'' Fenner, Gasarch, Glover, and Purewall consider the following: A two-dimensional grid is a set Gn,m = [n] x [m]. A grid Gn,m is c-colorable if there is a function χn,m : Gn,m→[c] such that there are no rectangles with all four corners the same color. This raises the natural question: for which values of n and m is Gn,m c-colorable? In "Monochromatic Boxes in Colored Grids,'' Cooper, Fenner, and Purewal give upper bounds for 3-dimensional grids with 2-colorings.In this work, we discuss Ramsey Theory and its various impact on mathematics and give specific colorings of grids for [3] x [7] x [126], [4] x [7] x [126], and [5] x [5] x [100], that provide lower bounds for the Ramsey-type numbers considered in Cooper et al.

ISBN: 9798819365564Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Combinatorics

Ramsey Theory and 2-Colorings of 3-Dimensional Grids.
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520 $a In 1928 the English mathematician Frank Plumpton Ramsey proved that patterns are actually implicit in any large structure. In fact, Ramsey theory states that any structure will necessarily contain an orderly substructure. As the late American mathematician Theodore S. Motzkin first proclaimed, Ramsey theory implies that complete disorder is an impossibility. In their paper "Rectangle Free Coloring of Grids,'' Fenner, Gasarch, Glover, and Purewall consider the following: A two-dimensional grid is a set Gn,m = [n] x [m]. A grid Gn,m is c-colorable if there is a function χn,m : Gn,m→[c] such that there are no rectangles with all four corners the same color. This raises the natural question: for which values of n and m is Gn,m c-colorable? In "Monochromatic Boxes in Colored Grids,'' Cooper, Fenner, and Purewal give upper bounds for 3-dimensional grids with 2-colorings.In this work, we discuss Ramsey Theory and its various impact on mathematics and give specific colorings of grids for [3] x [7] x [126], [4] x [7] x [126], and [5] x [5] x [100], that provide lower bounds for the Ramsey-type numbers considered in Cooper et al.
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