Recently, the DeepMind team from Google announced some astonishing news — their system AlphaEvolve has made significant progress in studying Ramsey numbers. These values are among the most complex objects in combinatorial mathematics and have remained practically inaccessible for precise calculations for many years.
To put it simply, Ramsey numbers, for example R(s,t), are the minimum number of vertices in a complete graph such that any two-coloring of its edges (for instance, red and blue) inevitably results in either a fully red subgraph on s vertices or a fully blue one on t vertices. In other words, if you have such a graph, it must contain this structure.
The problem is that calculating these numbers is incredibly difficult. Even for relatively small values like R(5,5), the exact value has remained unknown for nearly a century. The main challenge lies in the fact that determining these numbers requires examining an enormous number of coloring options — an exponential set that even supercomputers can hardly process entirely.
Legendary mathematician Erdős once joked that only aliens or future civilizations would be able to find the exact values for R(5,5). And all because this problem is truly very hard.
And then suddenly, DeepMind announced: their system AlphaEvolve not only confirmed the best existing estimates for Ramsey numbers in five classical cases but also improved them! That is, they obtained new lower bounds where results had been missing for decades.
What’s especially interesting is that before this achievement, everything relied on manual algorithms and clever techniques devised by humans. Now, a universal model based on large language models (LLMs) was able to independently find such solutions and approach the State of the Art.
This is truly impressive progress — nothing like this had been done even by the most powerful systems before.
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