The interface between theoretical mathematics and practical computation continues to evolve rapidly. While computers have become increasingly sophisticated tools for mathematical exploration and verification, discovering new mathematical constructions remains a delicate interplay between human insight and computational guidance. This is particularly challenging for problems that combine discrete choices with continuous aspects. At the Zuse Institute, we have developed a novel approach using neural net-works to guide mathematical intuition in precisely such settings, leading to the first improvement in thirty years for a variant of the famous Hadwiger–Nelson problem in geometric graph theory.
The Hadwiger–Nelson problem, first posed in 1950, asks a deceptively simple question: how many colors are needed to paint a plane so that no two points at a distance of exactly 1 from each other have the same color? More formally, we want to know the chromatic number of the plane χ(ℝ2), that is the smallest integer c for which there exists a function 𝑓 : ℝ2 → {1, . . . , c} such that 𝑓(𝓍) ≠ 𝑓(y) for any 𝓍, 𝓎 ∈ ℝ2 satisfying ∥𝓍 − 𝓎∥2 = 1.
Despite its elementary statement, this problem has resisted resolution for over 70 years. We know that at least five colors are needed, famously proved by De Grey in 2018, and that seven colors are always suffi-cient, shown through an explicit construction in 1950. One natural variant asks whether six colors might suffice if we allow the sixth color to avoid a different distance than one.
The key idea was to reformulate this geometric col-oring problem as a continuous optimization task that neural networks could tackle effectively. Rather than searching for a discrete coloring function 𝑓, we trained networks to output probability distributions 𝓅𝓍 ∈ [0, 1]c over c colors at each point 𝓍, minimizing the expected occurrence of forbidden patterns through a differentia-ble loss function ℒ(𝓅) = 𝔼∥𝓍−𝓎∥=1[𝓅𝓍 . 𝓅𝓎].
ble loss function ℒ(𝓅) = 𝔼∥𝓍−𝓎∥=1[𝓅𝓍 . 𝓅𝓎]. The results exceeded our expectations. The neural networks consistently suggested patterns that, after careful mathematical analysis and formalization, led to two novel six-colorings of the plane. These con-structions significantly expanded the known range of distances that could be avoided in the sixth color, extending it from the previously known interval of [0.415, 0.447] to [0.354, 0.657]. This represents the first improvement to these bounds since the work of Hoffman and Soifer in 1993.
Beyond these specific results, our work showcases a promising direction for AI-assisted mathematics. Rather than attempting fully automated proofs, we use neural networks to explore possible constructions and suggest patterns that human mathematicians can then analyze rigorously. This computer-guided intu-ition proved particularly valuable in our case, where the final constructions involved intricate geometric patterns that would have been difficult to discover through traditional methods.
Looking ahead, the framework we developed is not lim-ited to the Hadwiger–Nelson problem. Our approach has already shown promise in exploring related prob-lems, including variants involving monochromatic triangles with prescribed side lengths and colorings of three-dimensional space. More broadly, similar techniques could be applied to other mathematical do-mains where continuous relaxations are possible, such as using graphons for problems in extremal graph theory. The success of neural networks in suggesting novel mathematical constructions, as demonstrated by recent work across various domains, highlights the growing synergy between machine learning and pure mathematics. This synthesis of computational explo-ration and mathematical rigor represents a promising direction for tackling other long-standing open prob-lems in mathematics.
Konrad Mundinger, Max Zimmer, Christoph Spiegel, Sebastian Pokutta
