Long-Horizon AI Research for Grothendieck Constant: A Case Study in Human-AI Mathematical Collaboration

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.