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

在数学的分支组合学中，图是一个抽象的对象，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。我们说一个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标签（称为颜色）的规则，图的色数是为图着色所需的最小颜色数，使得没有两个相邻顶点是相同的颜色。图着色有许多实际应用。举个例子，假设你想在同一天邀请多个人参加面试，但是有几对候选人你不想同时面试。您需要的最小时隙数是多少？把候选人看作是一个图的顶点，如果要把他们放在不同的时隙中，就用一条边把他们连接起来。如果我们让不同的颜色代表不同的时隙，那么回答我们的问题就相当于确定图的色数。这仅仅是一个例子，展示了如何轻松地将一个普通的物流问题转化为一个关于图着色的问题。在生物学、物理学、工业、计算机科学、社会科学和媒体（例如社交网络）中存在许多类似的问题。我们的建议的一部分是确定这些问题，并使用我们的数学技术来解决它们。我们的方法来研究色数是通过一个意想不到的路线。我们将考虑重要的无限图族的色数，通过将它们与丢番图逼近（分数逼近数的研究）和动力系统中的问题联系起来。这是一个很有前途的新方向，它将推动当前数学知识的边界，并为许多学科之间的新思想流动打开大门。

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.