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

在称为组合学的数学分支中，图是一个抽象对象，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。如果有一条边将图中的两个顶点连接起来，我们就说它们相邻。图的着色是为每个顶点分配一个标签（称为颜色）的规则，图的色数是为图着色所需的最小颜色数，以便没有两个相邻的顶点是相同的颜色。图形着色有许多实际应用。举个例子，假设你想在同一天邀请一些人参加面试，但有一些候选人你不想同时面试。你需要的最小时隙数是多少？把候选点想象成一个图的顶点，如果要把它们放在不同的时隙里，就用一条边把它们连接起来。如果我们用不同的颜色表示不同的时隙，那么回答这个问题就等于确定了图的色数。这仅仅是一个例子，用来证明一个人可以很容易地把一个常见的物流问题变成一个关于图形着色的问题。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）中有许多这样的问题。我们的部分建议是找出这些问题，并用我们的数学技术来解决它们。我们研究色数的方法是通过一条意想不到的途径。我们将考虑无限图的重要族的色数，通过将它们与丢芬图近似（用分数逼近数的研究）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，它将突破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.