Long-Horizon AI Research for Grothendieck Constant: A Case Study in Human-AI Mathematical Collaboration
2026-08-11 • Artificial Intelligence
Artificial IntelligenceComputational ComplexityHuman-Computer Interaction
AI summaryⓘ
The authors studied how AI can help improve mathematical results by focusing on the Grothendieck constant, a number important in understanding some hard math problems. They managed to narrow down the possible values of this constant using an AI system that produced new ideas recognized as valuable by math experts. The paper also shares their experiences on what worked well and what didn’t when using AI to make these mathematical breakthroughs. Their work offers insights into effectively combining human expertise and AI in math research.
Grothendieck constantcombinatorial problemscontinuous relaxationboundsAI research systemmathematics researchnovel insightsmachine learningoptimization
Authors
Alan Li, Rahul Saha, Anton Xue, Swarat Chaudhuri, Adam Klivans, Pravesh K Kothari, Raghu Meka
Abstract
AI agents are increasingly used in mathematics research, but it is often unclear how to use them effectively. Towards this, we present an extensive case study of how AI was used to improve bounds on the Grothendieck constant $K_G$, which captures the hardness between combinatorial problems and their continuous relaxations. Specifically, while the precise value of $K_G$ is not known, we recently tightened the best known bounds to \[ \frac{6π}{11} \;\le\; K_G \;\le\; \fracπ{2\log(1+\sqrt2)} - 10^{-4}. \] Crucially, these improvements were achieved using an AI research system that could arrive at insights deemed novel by domain experts. We give a detailed discussion of our experience using AI for mathematics research, particularly touching upon its strengths and weaknesses, as well as our experience with creating ideal conditions for AI to arrive at breakthrough insights.