对角 Ramsey 数的上界 [继 Campos, Griffiths, Morris, 和 Sahasrabudhe 之后]
Upper bounds on diagonal Ramsey numbers [after Campos, Griffiths, Morris, and Sahasrabudhe]
AI总结：
本文综述了对角 Ramsey 数上界问题的历史，并介绍了 Campos 等人将上界改进至 $N=3.993^k$ 的突破性证明及后续的替代性概念证明。
摘要：
Ramsey 定理指出，如果 $N$ 足够大，那么无论人们如何用两种颜色对 $N$ 个顶点之间的边进行着色，总存在 $k$ 个顶点，它们所张成的边全为同一颜色。基于此定理，自然会产生疑问：“多大才算足够大？”Ramsey 最初的证明表明 $N=k!$ 就足够了，五年后 Erdős 和 Szekeres 将此上界改进为 $N=4^k$。随后，该问题的进展停滞了将近 90 年。在这篇综述中，我介绍了该问题的历史，并讨论了 Campos--Griffiths--Morris--Sahasrabudhe 近期突破中使用的一些思想，他们证明了 $N=3.993^k$ 就足够了。此外，我还讨论了 Balister, Bollobás, Campos, Griffiths, Hurley, Morris, Sahasrabudhe 和 Tiba 的后续工作，他们给出了一种替代且更具概念性的证明。
英文摘要：
Ramsey's theorem states that if $N$ is sufficiently large, then no matter how one colors the edges among $N$ vertices with two colors, there are always $k$ vertices spanning edges in only one color. Given this theorem, it is natural to ask ``how large is sufficiently large?'' Ramsey's original proof showed that $N=k!$ is sufficient, and five years later Erdős and Szekeres improved this bound to $N=4^k$. And then progress stalled for almost 90 years. In this survey, I present the history of the problem and discuss some of the ideas used in the recent breakthrough of Campos--Griffiths--Morris--Sahasrabudhe, who proved that $N=3.993^k$ is sufficient. In addition, I discuss the subsequent work of Balister, Bollobás, Campos, Griffiths, Hurley, Morris, Sahasrabudhe, and Tiba, who gave an alternative, and more conceptual, proof.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
