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「Open question: triangle and diamond densities in large dense graphs」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

The question in extremal graph theory I wish to discuss here originates from Luca Trevisan ; it shows that we still don’t know everything that we should about the “local” properties of large dense graphs.

Let be a large (undirected) graph, thus V is the vertex set with some large number n of vertices, and E is the collection of edges {x,y} connecting two vertices in the graph. We can allow the graph to have loops {x,x} if one wishes; it’s not terribly important for this question (since the number of loops is so small compared to the total number of edges), so let’s say there are no loops. We define three quantities of the graph G:

已知结果和反例

Up to insignificant errors of o(1) (i.e. anything which goes to zero as the number of vertices goes to infinity), these densities can also be interpreted probabilistically as follows: if x,y,z,w are randomly selected vertices in V, then

(The errors of o(1) arise because the vertices x,y,z,w may occasionally collide with each other, though this probability becomes very small when n is large.) Thus we see that these densities are “local” qualities of the graph, as we only need to statistically sample the graph at a small number of randomly chosen vertices in order to estimate them.

证明或构造的主线

A general question is to determine all the constraints relating in the limit . (It is known from the work of Lovász and Szegedy that the relationships between local graph densities such as these stabilise in this limit; indeed, given any error tolerance and any large graph G with densities , there exists a graph with “only” vertices whose densities differ from those of G by at most , although the best known bounds for are far too poor at present to be able to get any useful i

Let us forget about diamonds for now and only look at the edge and triangle densities . Then the story is already rather non-trivial. The main concern is to figure out, for each fixed , what the best possible upper and lower bounds on are (up to o(1) errors); since the collection of graphs with a given edge density is “path-connected” in some sense, it is not hard to see that every value of between the upper and lower bounds is feasible modulo o(1) errors.

阅读时建议盯住的点

The best possible upper bound is easy: . This can be established by either the Kruskal-Katona theorem , the Loomis-Whitney inequality (or the closely related box theorem ), or just two applications of Hölder’s inequality ; we leave this as an exercise. The bound is sharp, as can be seen by looking at a complete subgraph on vertices. (We thank Tim Austin and Imre Leader for these observations and references, as well as those in the paragraph below.) [ Update , Apr 21: There is

The lower bound is trickier. The complete bipartite graph example shows that the trivial lower bound is attainable when , and Turán’s theorem shows that this is sharp. For , a classical theorem of Goodman (see also Nordhaus and Stewart ) shows that . When for some integer k, this inequality is sharp, as can be seen by looking at the complete k-partite graph.

值得单独记下的条目

• The edge density , defined as the number of edges in G, divided by the total number of possible edges, i.e. ;
• The triangle density , defined as the number of triangles in G (i.e. unordered triplets {x,y,z} such that {x,y},{y,z}, {z,x} all lie in G), divided by the total number of possible triangles, namely ;
• The diamond density , defined as the number of diamonds in G (i.e. unordered pairs { {x,y,z}, {x,y,w} } of triangles in G which share a common edge), divided by the total number of possible diamonds, namely .

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。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源