陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「The crossing number inequality」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

Today I’d like to discuss a beautiful inequality in graph theory, namely the crossing number inequality . This inequality gives a useful bound on how far a given graph is from being planar, and has a number of applications, for instance to sum-product estimates. Its proof is also an excellent example of the amplification trick in action; here the main source of amplification is the freedom to pass to subobjects, which is a freedom which I didn’t touch upon in the previous pos

在这类讨论里, when I talk about a graph , I mean an abstract collection of vertices V, together with some abstract edges E joining pairs of vertices together. We will assume that the graph is undirected (the edges do not have a preferred orientation), loop-free (an edge cannot begin and start at the same vertex), and multiplicity-free (any pair of vertices is joined by at most one edge). More formally, we can model all this by viewing E as a subset of , the set of 2-element subsets

已知结果和反例

Now one of the great features of graphs, as opposed to some other abstract maths concepts, is that they are easy to draw: the abstract vertices become dots on a plane, while the edges become line segments or curves connecting these dots. [To avoid some technicalities we do not allow these curves to pass through the dots, except if the curve is terminating at that dot.] Let us informally refer to such a concrete representation D of a graph G as a drawing of that graph. Clearly

Given an abstract graph G, or a drawing thereof, it is not always obvious as to whether that graph is planar; just because the drawing that you currently possess of G contains crossings, does not necessarily mean that all drawings of G do. The wonderful little web game “ Planarity ” illustrates this point excellently. Nevertheless, there are definitely graphs which are not planar; in particular the complete graph on five vertices, and the complete bipartite graph on two sets

证明或构造的主线

There is in fact a famous theorem of Kuratowski that says that these two graphs are the only “source” of non-planarity, in the sense that any non-planar graph contains (a subdivision of) one of these graphs as a subgraph. (There is of course the even more famous four-colour theorem that asserts that every planar graphs is four-colourable, but this is not the topic of my post today.)

Intuitively, if we fix the number of vertices |V|, and increase the number of edges |E|, then the graph should become “increasingly non-planar”; conversely, if we keep the same number of edges |E| but spread them amongst a greater number of vertices |V|, then the graph should become “increasingly planar”. Is there a quantitative way to measure the “non-planarity” of a graph, and to formalise the above intuition as some sort of inequality?

阅读时建议盯住的点

It turns out that there is an elegant inequality that does precisely this, known as the crossing number inequality . It was first discovered by Ajtai-Chvátal-Newborn-Szemerédi and by Leighton (the latter being motivated by optimising VLSI designs). Nowadays it can be proven by two elegant amplifications of Euler’s formula , as we shall see.

If D is a drawing of a graph G, we define cr(D) to be the total number of crossings – where pairs of edges intersect at a point, for a reason other than sharing a common vertex. (If multiple edges intersect at the same point, each pair of edges counts once.) We then define the crossing number cr(G) of G to be the minimal value of cr(D) as D ranges over the drawings of G. Thus for instance cr(G)=0 if and only if G is planar. One can also verify that the two graphs and have a c

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

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