Diophantine approximation, chromatic number, and equivalence classes of separated nets

在被称为组合学的数学分支中，图是一个抽象对象，它可以被认为是点(称为顶点)的集合，其中一些点由线段(称为边)连接。我们说图中的两个顶点是相邻的，如果有一条边连接它们。图的着色是为每个顶点分配一个标号(称为颜色)的规则，而图的色数是使相邻的两个顶点都不是相同颜色所必需的最小色数。图的着色有许多实际应用。例如，假设你想在同一天邀请许多人参加面试，但有几对候选人你不想在同一时间面试他们。您需要的最低时隙数是多少？将候选对象想象成图的顶点，如果要将它们放在不同的时隙中，则有一条边相互连接。如果我们用不同的颜色表示不同的时隙，那么回答我们的问题就相当于确定图的色数。这只是一个例子，说明人们可以很容易地将一个常见的物流问题转化为关于图着色的问题。在生物学、物理学、工业、计算机科学以及社会科学和媒体(例如社交网络)中，有很多这样的问题。我们建议的一部分是识别这些问题并使用我们的数学技巧来解决它们。我们研究色数的方法是通过一条意想不到的途径。我们将通过将它们与丢番图逼近(分数逼近的研究)和动力系统中的问题联系起来，来考虑重要的无限图族的色数。这是一个很有希望的新方向，它将推动数学现有知识的边界，并为许多学科之间的新思想交流打开大门。

In the branch of mathematics called combinatorics, a graph is an abstract object which can be thought of as a collection of points (called vertices), some of which are connected by line segments (called edges). We say that two vertices in a graph are adjacent if there in an edge connecting them. A colouring of a graph is a rule that assigns a label (called a colour) to each vertex, and the chromatic number of the graph is the minimum number of colours necessary to colour the graph so that no two adjacent vertices are the same colour.Graph colourings have a multitude of practical applications. As an example, suppose you would like to invite a number of people for interviews on the same day but that there are certain pairs of candidates whom you don't want to interview at the same time. What is the minimum number of time slots that you need? Think of the candidates as the vertices of a graph, with an edge connecting one to another if they are to be put in different time slots. If we let our different colours represent different time slots, then answering our question is equivalent to determining the chromatic number of the graph. This is merely an example to demonstrate the ease with which one can turn a common logistics problem into a problem about graph colourings. There are many problems like this in biology, physics, industry, computer science, and in the social sciences and media (for example social networking). Part of our proposal is to identify these problems and to use our mathematical techniques to solve them.Our approach to studying chromatic number is via an unexpected route. We will be considering the chromatic number of important families of infinite graphs, by connecting them with problems in Diophantine approximation (the study of approximation of numbers by fractions) and dynamical systems. This is a promising new direction which will push the boundary of current knowledge in mathematics and open the door for the flow of new ideas between many subjects.