After 80 Years, Mathematicians Give Famed ‘Erdős Method’ an Upgrade

In 1947, Paul Erdős, the itinerant Hungarian mathematician, introduced what would become one of math’s most powerful tools.

1947 年，匈牙利巡回数学家 Paul Erdős 推出了后来成为最强大的数学工具之一的东西。

He wanted to prove that a certain kind of object existed — in this case, a network made of interconnected nodes.

他想证明某种物体的存在——在这种情况下，是一个由互连节点组成的网络。

But strangely, his proof didn’t specify how to build it.

但奇怪的是，他的证明并没有具体说明如何构建它。

Instead, he showed that if you consider all networks and select one at random, the chances that you’ll find a network with the property you want is greater than zero.

相反，他表明，如果你考虑所有网络并随机选择一个，那么你找到具有你想要的属性的网络的机会大于零。

That means that the desired network is out there somewhere, even if you know almost nothing about it.

这意味着所需的网络就在某个地方，即使您对此几乎一无所知。

Erdős’ approach, known as the probabilistic method, was simple but revolutionary.

埃尔多斯的方法被称为概率方法，简单但具有革命性。

Before its development, “if I’m telling you that certain objects exist, you would tell me, ‘Show me,’” said Benny Sudakov , a mathematician at the Swiss Federal Institute of Technology Zurich.

在它开发之前，“如果我告诉你某些物体存在，你会告诉我，‘给我看’，”瑞士苏黎世联邦理工学院的数学家本尼·苏达科夫说。

“But certain objects are so unusual that it’s hard for us to grasp that they exist at all.”

“但某些物体是如此不寻常，以至于我们很难理解它们的存在。”

Erdős’ technique overcame this difficulty, demonstrating that randomness could be used in ways mathematicians had never imagined.

埃尔多斯的技术克服了这一困难，证明随机性可以以数学家从未想象过的方式使用。

“It was just astounding that you would use randomness,” said Joel Spencer of New York University.

纽约大学的乔尔·斯宾塞 (Joel Spencer) 表示：“使用随机性真是令人震惊。”

“Now, that’s the baseline.”

“现在，这就是底线。”

Today, the probabilistic method is used across mathematics and computer science — to figure out if a number is prime, to design better circuits, or to clean up data without introducing biases.

如今，概率方法在数学和计算机科学中得到广泛应用——确定一个数是否为素数，设计更好的电路，或者在不引入偏差的情况下清理数据。

Researchers have strengthened the technique in various ways.

研究人员以各种方式加强了这项技术。

But the original focus of the probabilistic method — the question about networks that Erdős sought to answer — has seen very little progress.

但概率方法最初的关注点——埃尔多斯试图回答的有关网络的问题——却几乎没有取得任何进展。

For eight decades, mathematicians were unable to significantly improve on the solution that Erdős came up with.

八十年来，数学家们无法对埃尔多斯提出的解决方案进行显着改进。

That’s now finally starting to change.

现在这种情况终于开始改变。

Imagine a network of nodes — a graph — in which every pair of nodes is connected by an edge.

想象一个节点网络（一个图），其中每对节点都通过一条边连接。

Now color each edge either red or blue, but with one caveat: Don’t create any large clusters of nodes that are all connected by edges of the same color.

现在将每条边着色为红色或蓝色，但有一个警告：不要创建任何由相同颜色的边连接的大型节点簇。

These forbidden structures are called monochromatic cliques.

这些禁止的结构被称为单色团。

Here’s a monochromatic clique consisting of three nodes, which mathematicians call a clique of size 3:

这是一个由三个节点组成的单色团，数学家将其称为大小为 3 的团：

If your graph has enough nodes, it will be impossible to avoid creating a monochromatic clique, no matter how you color the edges.

如果您的图形有足够的节点，则无论您如何为边缘着色，都将不可避免地创建单色团。

For instance, if you want to avoid a clique of size 3, your graph can have at most five nodes.

例如，如果您想避免规模为 3 的派系，则您的图表最多可以有五个节点。

A six-node graph will always have one:

六节点图总是有一个：

Mathematicians therefore say that the “Ramsey number” for a clique of size 3, denoted R (3), is 6.

因此，数学家说，规模为 3 的集团的“拉姆齐数”（表示为 R (3)）为 6。

Ramsey numbers measure how big graphs can get before the forbidden pattern inevitably emerges.

拉姆齐数衡量了在禁止模式不可避免地出现之前图表可以达到多大。

You can also have Ramsey numbers for red and blue cliques of different sizes.

您还可以拥有不同大小的红色和蓝色派系的拉姆齐数。

For example, you can color an eight-node graph so that it has no red cliques of size 3 or blue cliques of size 4.

例如，您可以对八节点图进行着色，使其没有大小为 3 的红色团或大小为 4 的蓝色团。

But if you add one more node to your graph, you will be forced to create at least one red or blue clique.

但是，如果您在图表中再添加一个节点，您将被迫创建至少一个红色或蓝色团。

Therefore, the Ramsey number R (3, 4) is 9.

因此，拉姆齐数 R (3, 4) 为 9。

As the cliques you want to avoid get bigger, the problem gets more and more difficult to solve.

当你想要避免的派系变得越来越大时，问题就会变得越来越难以解决。

Mathematicians have been able to calculate only a handful of the smallest Ramsey numbers.

数学家只能计算出少数最小的拉姆齐数。

“It’s very hard to create something that has no structure,” said Paul Horn of the University of Denver.

丹佛大学的保罗·霍恩说：“创造没有结构的东西非常困难。”

“Maybe it’s because we’re human and we’re subject to our biases.”

“也许这是因为我们是人类，我们会受到偏见的影响。”

And so mathematicians have spent decades trying to find better and better approximations of Ramsey numbers.

因此，数学家花了几十年的时间试图找到越来越好的拉姆齐数近似值。

That’s what Erdős was trying to do when he introduced his probabilistic method in 1947.

这就是 Erdős 在 1947 年引入概率方法时试图做的事情。

Instead of building clique-free graphs directly, he considered every possible way to color a graph, then showed that at least some nonzero fraction of them must be clique-free.

他没有直接构建无派系图，而是考虑了为图着色的所有可能方法，然后表明其中至少有一些非零部分必须是无派系图。

Erdős used this argument to prove that, if you forbid red and blue cliques of size k , the Ramsey number R ( k ) must be bigger than $latex \sqrt{2}^k$ .

Erdős 使用这个论点来证明，如果禁止大小为 k 的红色和蓝色团，则拉姆齐数 R ( k ) 必须大于 $latex \sqrt{2}^k$ 。

Ramsey numbers for same-size red and blue cliques are called diagonal Ramsey numbers.

相同大小的红色和蓝色团的拉姆齐数称为对角拉姆齐数。

Erdős could similarly get a lower bound on “off-diagonal” Ramsey numbers R ( k , l ), in which you forbid red cliques of size k and blue cliques of size l .

Erdős 可以类似地得到“非对角线”拉姆齐数 R ( k , l ) 的下界，其中禁止大小为 k 的红色团和大小为 l 的蓝色团。

The proof was just a few lines long.

证明只有几行长。

But it was completely unexpected.

但这是完全出乎意料的。

At first, mathematicians were loath to follow his lead.

起初，数学家们不愿意追随他的脚步。

They wanted concrete examples.

他们想要具体的例子。

“For many years, Erdős was like a voice in the wilderness,” Spencer said.

“多年来，埃尔多斯就像荒野中的一个声音，”斯宾塞说。

“He was getting these amazing results using randomness, and people had never done that before.”

“他利用随机性得到了这些惊人的结果，人们以前从未这样做过。”

But soon the probabilistic method proved its worth.

但很快概率方法就证明了它的价值。

It’s now one of the most ubiquitous techniques in “discrete” math, the study of objects (like graphs) that are separate rather than continuous.

它现在是“离散”数学中最普遍的技术之一，即研究分离而不是连续的对象（例如图形）。

And it has seeped out of math into physics and computer science .

它已经从数学渗透到物理学和计算机科学中。

“The randomness, I think, just helps us get at something that is otherwise very ethereal,” Horn said.

“我认为，随机性只会帮助我们获得一些原本非常空灵的东西，”霍恩说。

More recently, mathematicians have been able to adapt Erdős’ method to get better estimates of Ramsey numbers where the forbidden cliques differ vastly in size .

最近，数学家们已经能够采用 Erdős 的方法来更好地估计拉姆齐数，其中禁止集团的大小差异很大。

For instance, in 2025 Horn and three colleagues used an updated version of Erdős’ method to prove a more precise lower bound for R (3, l ) , where l grows arbitrarily large.

例如，2025 年，Horn 和三位同事使用 Erdős 方法的更新版本来证明 R (3, l ) 的更精确下界，其中 l 增长任意大。

(That work, in turn, led to a major breakthrough in graph theory.)

（这项工作反过来又导致了图论的重大突破。）

Paul Erdős figured out how to use randomness to prove that certain mathematical objects exist, even if you don’t know how to construct them.

Paul Erdős 弄清楚了如何利用随机性来证明某些数学对象的存在，即使你不知道如何构造它们。

His technique, today known as the probabilistic method, transformed many branches of math and computer science.

他的技术，今天被称为概率方法，改变了数学和计算机科学的许多分支。

Archives of the Mathematisches Forschungsinstitut Oberwolfach ©Gabriella Bollobas

Oberwolfach 数学研究所档案 ©Gabriella Bollobas

But when it came to Ramsey numbers where the forbidden cliques weren’t so different in size — particularly diagonal Ramsey numbers, the object of Erdős’ original interest — the probabilistic method stalled.

但当涉及拉姆齐数时，禁止团的大小差异不大——特别是对角拉姆齐数，这是埃尔多斯最初感兴趣的对象——概率方法就停滞了。

Say you forbid cliques of size 1,000.

假设您禁止 1,000 人的派系。

Erdős showed that R (1,000) must be bigger than about 2 500 .

Erdős 表明 R (1,000) 必须大于约 2 500 。

Eight decades of effort changed that bound to about 2 501 .

经过八年的努力，这个界限变成了大约 2 501 。

Similarly, from the 1970s onward, progress remained stock-still for off-diagonal Ramsey numbers where the forbidden red and blue cliques are both relatively large.

同样，从 20 世纪 70 年代开始，非对角拉姆齐数的进展仍然停滞不前，其中禁止的红色和蓝色团都相对较大。

Then along came a graduate student with barely any expertise in Ramsey theory.

然后出现了一位对拉姆齐理论几乎没有任何专业知识的研究生。

Wujie Shen had spent his first few semesters at Tsinghua University focused mainly on geometry and topology.

沉无杰在清华大学的前几个学期主要学习几何和拓扑。

But in the spring of 2024, he came across a paper on Ramsey numbers that captivated him.

但在 2024 年春天，他发现了一篇关于拉姆齐数字的论文，这让他着迷。

He knew how Erdős’ method worked: You flip a coin to determine the color of each edge of your graph: Heads, the edge is red; tails, it’s blue.

他知道埃尔多斯的方法是如何运作的：你抛一枚硬币来确定图表每条边的颜色：正面，边缘是红色；正面，边缘是红色；正面，边缘是红色。尾巴，是蓝色的。

You then calculate the probability that you’ll get a clique-free coloring.

然后，您可以计算获得无派系着色的概率。

But this calculation gets very difficult for larger graphs.

但对于较大的图来说，这种计算变得非常困难。

Shen wondered whether there was a random model that could produce clique-free colorings more efficiently than Erdős’ approach.

Shen 想知道是否存在一种随机模型能够比 Erdős 的方法更有效地产生无派系着色。

Given Shen’s training, it’s perhaps no surprise that the model he came up with involved geometry.

鉴于沉的训练，他提出的模型涉及几何也许并不奇怪。

Typically, graph colorings don’t invoke geometry: All that matters to mathematicians is which nodes are connected by a red edge, and which are connected by a blue one.

通常，图形着色不会调用几何图形：对数学家来说，重要的是哪些节点通过红色边缘连接，哪些节点通过蓝色边缘连接。

Whether those nodes sit close together or are scattered throughout space has no significance.

这些节点是否靠近在一起或分散在整个空间中并不重要。

But Shen wanted to use geometry to help him decide which edges to color red and which to color blue.

但 Shen 希望利用几何学来帮助他决定将哪些边涂成红色，将哪些边涂成蓝色。

In particular, he wanted to use the geometry of high-dimensional spheres — that is, sets of points that are equidistant from a single central point.

特别是，他想使用高维球体的几何形状，即距单个中心点等距的点集。

These spheres “mess with all our intuitions completely,” said David Conlon of the California Institute of Technology.

加州理工学院的大卫·康伦说，这些球体“完全扰乱了我们的直觉”。

Many of our assumptions about what a sphere looks like are no longer true in high dimensions: A high-dimensional sphere has a tiny volume and massive surface area, and most of its points lie on the equator.

我们关于球体外观的许多假设在高维中不再正确：高维球体具有很小的体积和巨大的表面积，并且它的大部分点位于赤道上。

It’s “pretty complicated to work with,” Sudakov said.

苏达科夫说，它“使用起来相当复杂”。

But Shen and two colleagues — Jie Ma , who was visiting Tsinghua to teach for the fall term, and Ma’s graduate student Shengjie Xie — wanted to try.

但沉和两位同事——秋季学期来清华大学任教的马杰，以及马杰的研究生谢胜杰——想尝试一下。

Their method: First, place nodes one by one onto the surface of a high-dimensional sphere.

他们的方法是：首先，将节点逐个放置到高维球体的表面上。

Choose each node’s position at random — any point on the sphere is fair game, and the placement of each node has no influence over the placement of any other node.

随机选择每个节点的位置 - 球体上的任何点都是公平的游戏，每个节点的位置不会影响任何其他节点的位置。

Once you’ve placed all the nodes, color each edge based on the distance between the nodes.

放置完所有节点后，根据节点之间的距离为每条边着色。

If two points are more than some fixed distance apart (which will happen with a probability of less than 1/2), color the edge connecting them red.

如果两个点之间的距离超过某个固定距离（发生这种情况的概率小于 1/2），则将连接它们的边涂成红色。

If they’re closer together, color the edge blue.

如果它们靠得更近，请将边缘涂成蓝色。

From left/top: Jie Ma, Wujie Shen, and Shengjie Xie used the strange geometry of high-dimensional spheres to make progress on a problem that had been stalled for decades.

从左/上起：马杰、沉五杰和谢胜杰利用高维球体的奇怪几何形状，在一个停滞了几十年的问题上取得了进展。

From left/top: Courtesy of Jie Ma; Courtesy of Wujie Shen; Ziyuan Zhao

从左/上：由马杰提供；由沉无杰提供；赵子元

With this approach, the graphs that Ma, Shen, and Xie created were less likely to form a red clique.

通过这种方法，马、沉和谢创建的图表不太可能形成红色派系。

That’s because to form a large red clique, you need many nodes that are all far away from one another.

这是因为要形成一个大的红色派系，你需要许多彼此相距较远的节点。

With only so much space on the sphere, this is unlikely to happen.

由于球体上的空间有限，这种情况不太可能发生。

But there’s a catch.

但有一个问题。

By the same token, this method also produces a greater fraction of colorings that have blue cliques than Erdős’ does.

出于同样的原因，与 Erdős 的方法相比，该方法产生的具有蓝色团的颜色比例也更高。

“There’s a trade-off that looks like it really helps in one color, but it doesn’t help at all in the other color,” Conlon said.

“有一个权衡，看起来它确实对一种颜色有帮助，但对另一种颜色却毫无帮助，”康伦说。

“Why bother?”

“何苦呢？”

Even so, Ma, Shen, and Xie were hopeful.

即便如此，马、沉和谢仍抱有希望。

They tested their method on smaller graphs, and it seemed to work: Among the tens of thousands of bad colorings it generated, there was still a nonzero chance of getting a good clique-free coloring as well.

他们在较小的图表上测试了他们的方法，它似乎有效：在它生成的数以万计的不良着色中，仍然有非零的机会获得良好的无派系着色。

That reassured them that the benefits could outweigh the costs, even for much bigger graphs.

这让他们放心，即使对于更大的图表，收益也可能超过成本。

They then set out to prove it.

然后他们开始证明这一点。

The key turned out to be the very weird geometry of high-dimensional spheres.

事实证明，关键是高维球体非常奇怪的几何形状。

Ultimately, to show that they could avoid cliques of a particular size, Ma, Shen, and Xie needed to limit the probability that their randomly placed nodes formed clusters that were all far apart, or all close together.

最终，为了证明他们可以避免特定规模的派系，马、沉和谢需要限制他们随机放置的节点形成距离很远或靠得很近的簇的可能性。

They realized that if they drew lines from each node to the sphere’s center, those lines would almost all be perpendicular or close to perpendicular.

他们意识到，如果从每个节点到球体中心画线，这些线几乎都是垂直或接近垂直的。

That doesn’t happen if you randomly place nodes on a familiar two-dimensional sphere: Most nodes will not lie on perpendicular lines.

如果您将节点随机放置在熟悉的二维球体上，则不会发生这种情况：大多数节点不会位于垂直线上。

But the team was able to prove that it was true in the much higher dimensions that they were working in.

但该团队能够证明在他们工作的更高维度上这是正确的。

That, in turn, restricted how far nodes could be from one another — thereby limiting their chances of forming a monochromatic clique.

这反过来又限制了节点彼此之间的距离，从而限制了它们形成单色派系的机会。

After a year and 40 pages of dense computations, the trio posted their paper in July 2025.

经过一年时间和 40 页的密集计算，三人于 2025 年 7 月发表了他们的论文。

They’d improved Erdős’ lower bound on Ramsey numbers — but only when the forbidden blue cliques are larger than the red ones.

他们改进了埃尔多斯对拉姆齐数的下限——但前提是禁止的蓝色派系大于红色派系。

When the blue cliques are just as small as the red ones, the benefits of the new approach disappear.

当蓝色派系与红色派系一样小时，新方法的好处就会消失。

Still, when you want to avoid red cliques that are, say, half as large as blue ones, Ma, Shen, and Xie managed to nudge Erdős’ growth rate of $latex ((\sqrt{5} + 1)/2)^k$ up to $latex ((\sqrt{5} + 1)/2 + 10^{-21})^k$ .

尽管如此，当你想避免红色派系（例如，蓝色派系一半大）时，Ma、Shen 和 Xie 设法将 Erdős 的 $latex ((\sqrt{5} + 1)/2)^k$ 增长率提高到 $latex ((\sqrt{5} + 1)/2 + 10^{-21})^k$ 。

While the change is tiny, their proof marks the first improvement for near-diagonal Ramsey numbers in 50 years.

虽然变化很小，但他们的证明标志着近对角拉姆齐数 50 年来的首次改进。

“It’s lucky, and we feel like all our efforts are rewarded,” Ma said.

马云说：“很幸运，我们觉得所有的努力都得到了回报。”

“But it was tough for a long time.”

“但很长一段时间都是艰难的。”

“It’s a bit shocking that a familiar thing works for a familiar problem,” said Julian Sahasrabudhe of the University of Cambridge.

剑桥大学的朱利安·萨哈斯拉布德 (Julian Sahasrabudhe) 表示：“用熟悉的东西解决熟悉的问题，这有点令人震惊。”

Their technique, he said, “was hidden in plain view.”

他说，他们的技术“隐藏在众目睽睽之下”。

Ma, Shen, and Xie’s proof has already generated a spate of further progress.

马、沉和谢的证明已经取得了一系列进一步的进展。

In December 2025, Sudakov and two of his graduate students drastically simplified the team’s coloring model , improving their new bounds even further.

2025 年 12 月，苏达科夫和他的两名研究生极大地简化了团队的着色模型，进一步提高了他们的新界限。

Others have since used the model to estimate Ramsey numbers that involve three colors, not two.

此后，其他人使用该模型来估计涉及三种颜色而不是两种颜色的拉姆齐数。

That’s in keeping with the probabilistic method’s long history.

这与概率方法的悠久历史是一致的。

For the past 80 years, mathematicians have been tinkering with Erdős’ randomness-based technique, finding more and more ways to mix in additional structure to boost its power.

在过去的 80 年里，数学家们一直在修改 Erdős 基于随机性的技术，找到越来越多的方法来混合额外的结构以增强其威力。

Inevitably, these new techniques have then proved useful elsewhere.

不可避免地，这些新技术在其他地方被证明是有用的。

“It’s a very fruitful playground for ideas,” Sudakov said.

“这是一个富有成果的创意游乐场，”苏达科夫说。

Ma, Shen, and Xie’s work, then, is the latest chapter in this decades-old story.

那么，马、沉和谢的工作是这个长达数十年的故事的最新篇章。

But it’s also the first one in a long time to revisit the near-diagonal Ramsey numbers.

但这也是很长一段时间以来第一个重新审视近对角拉姆齐数的例子。

The team’s new contribution — a geometric approach — might lead to more progress on that stubborn problem.

该团队的新贡献——几何方法——可能会在这个顽固问题上取得更多进展。

Although the probabilistic method hasn’t been perfected yet, “it’s really very powerful now,” Spencer said.

尽管概率方法尚未完善，但“它现在确实非常强大，”斯宾塞说。

“It’s really changed so much.”

“变化真的太大了。”

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获取发送到您的电子邮件收件箱的最重要新闻的摘要

Quanta Magazine moderates comments to facilitate an informed, substantive, civil conversation.

《Quanta》杂志对评论进行审核，以促进知情、实质性、文明的对话。

Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected.

辱骂、亵渎、自我推销、误导、语无伦次或偏离主题的评论将被拒绝。

Moderators are staffed during regular business hours (New York time) and can only accept comments written in English.

版主在正常工作时间（纽约时间）有工作人员，并且只能接受英文评论。