然而，在涉及对角拉姆齐数（即避免特定大小的单色团的网络规模限制）的问题上，机率方法在过去八十年中进展极为有限。直到2025年，清华大学的沈吾介、马杰和谢盛杰利用高维球体的奇特几何性质，成功改进了近对角拉姆齐数的下界估计，实现了该领域五十年来的首次重大突破。
这一几何机率突破随后引发了该领域的连锁进展，例如数学家苏达科夫及其学生进一步简化了该模型并提高了估算界限。这项新成果不仅展示了几何方法的威力，也再次证明了在随机性中融入额外结构以增强其效果的强大潜力。
In 1947, Hungarian mathematician Paul Erdős introduced the probabilistic method, using randomness to prove the existence of specific network structures without providing a concrete construction. Today, this revolutionary tool is widely applied across mathematics and computer science, such as in primality testing, circuit design, and data cleaning.
However, progress on diagonal Ramsey numbers—which measure the size limits of networks that avoid monochromatic cliques of certain sizes—had remained stagnant for nearly eight decades. In 2025, Tsinghua University researchers Wujie Shen, Jie Ma, and Shengjie Xie successfully improved the lower bounds for near-diagonal Ramsey numbers by leveraging the unique geometric properties of high-dimensional spheres, marking the first major breakthrough in fifty years.
This geometric probabilistic breakthrough has since sparked consecutive advancements, such as mathematician Benny Sudakov and his students further simplifying the model and tightening the bounds. The success of this new work not only demonstrates the power of geometric approaches but also highlights the potential of mixing additional structure with randomness to solve complex mathematical problems.