OpenAI “internal model” solved 3 more Erdős problems

OpenAI has revealed that an internal, unreleased AI model has successfully solved three combinatorial mathematics problems that had remained open from the collection of famed mathematician Paul Erdős. The achievement was announced by researchers and confirmed by OpenAI's VP of Product, Kevin Weil, who noted the model's performance was "pretty wild." The work is detailed in a new paper titled "Solving Unsolved Problems in Combinatorics with Reinforcement Learning," which outlines a system that autonomously generates formal, machine-verifiable proofs.

The model employs a novel reinforcement learning approach, described as "AlphaProof-style," which combines a language model with a formal theorem prover. It was trained to navigate the complex logical space of combinatorics, specifically targeting problems involving sets and integers. Unlike previous AI systems that might find numerical solutions, this model's output is a complete, step-by-step logical proof that can be verified by existing proof-checking software. This represents a significant leap from pattern recognition to genuine deductive reasoning within a constrained domain.

While the specific internal model was not named (it is not GPT-4o or o1), the breakthrough underscores a major strategic push at OpenAI toward developing AI systems capable of deep, reliable reasoning. Automated theorem proving is considered a grand challenge in AI, as it requires planning, abstraction, and rigorous logical deduction—skills foundational to general intelligence. The success in a field as precise and unforgiving as combinatorics suggests the underlying techniques could eventually be applied to software verification, advanced scientific discovery, and more robust AI safety research.