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How Does One Cut a Triangle? by Alexander Soifer Pdf
This second edition of Alexander Soifer’s How Does One Cut a Triangle? demonstrates how different areas of mathematics can be juxtaposed in the solution of a given problem. The author employs geometry, algebra, trigonometry, linear algebra, and rings to develop a miniature model of mathematical research.
The Triangle-Free Process and the Ramsey Number R(3,k) by Gonzalo Fiz Pontiveros,Simon Griffiths,Robert Morris Pdf
The areas of Ramsey theory and random graphs have been closely linked ever since Erdős's famous proof in 1947 that the “diagonal” Ramsey numbers R(k) grow exponentially in k. In the early 1990s, the triangle-free process was introduced as a model which might potentially provide good lower bounds for the “off-diagonal” Ramsey numbers R(3,k). In this model, edges of Kn are introduced one-by-one at random and added to the graph if they do not create a triangle; the resulting final (random) graph is denoted Gn,△. In 2009, Bohman succeeded in following this process for a positive fraction of its duration, and thus obtained a second proof of Kim's celebrated result that R(3,k)=Θ(k2/logk). In this paper the authors improve the results of both Bohman and Kim and follow the triangle-free process all the way to its asymptotic end.
Author : Bruce K. Alexander,Curtis P. Shelton Publisher : Cambridge University Press Page : 563 pages File Size : 54,8 Mb Release : 2014-07-03 Category : History ISBN : 9781107007291
A Sharp Threshold for Random Graphs with a Monochromatic Triangle in Every Edge Coloring by Ehud Friedgut,Vojtěch Rödl,Andrzej Ruciński,Prasad Tetali Pdf
Let $\cal{R}$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let $G(n,p)$ be the random graph on $n$ vertices with edge probability $p$. We prove that there exists a function $\widehat c=\widehat c(n)=\Theta(1)$ such that for any $\varepsilon > 0$, as $n$ tends to infinity, $Pr\left[G(n,(1-\varepsilon)\widehat c/\sqrt{n}) \in \cal{R} \right] \rightarrow 0$ and $Pr \left[G(n,(1+\varepsilon)\widehat c/\sqrt{n}) \in \cal{R}\ \right] \rightarrow 1. A crucial tool that is used in the proof and is of independent interest is a generalization of Szemeredi's Regularity Lemma to a certain hypergraph setting.
Mathematics with Understanding by Harold Fletcher,Arnold A. Howell Pdf
Mathematics with Understanding: Book 2 is an eight-chapter book that begins with an explanation of the number systems. Fractions, rational numbers, systems of integers, algebraic relations, shape, and measurement are then described. The modular arithmetic and groups as well as probability are also explained. This book will be useful to students and teachers for a deeper understanding of the structure behind many mathematical ideas and processes.
Amazingly Simple Triangle Stars by Barbara H. Cline Pdf
Turn a single block into eight distinctive quilts—the magic is in the fabric! Best-selling author and well-known teacher Barbara H. Cline takes an easy-to-sew triangle block and then shows countless intricate designs for you to experiment with color and value. Stunning triangle-star quilts are perfect for the adventurous beginner looking to try something new or the intermediate quilter who wants to put her skills to use in a creative way. Design your own quilt top using a provided line drawing that you can color in and then bring to life with creative fabric selection!