Doubly Saturated Ramsey Graphs: A Case Study in Computer-Assisted Mathematical Discovery

There are several methods for searching for graphs with prescribed properties, such as SAT solvers and specialized generators. These methods return the result as raw data: an adjacency matrix or a string encoding. The raw data certifies that the graph exists, but it does not reveal any structural properties of the graph. We ask whether one can automatically discover a short algebraic description if only this raw data is provided. We look for a description such as a Cayley graph $\mathrm{Cay}(Γ, S)$ or a lexicographic product $C_5[K_3]$. We address this question with a neurosymbolic approach. We propose an agent that runs on a general-purpose large language model with no fine-tuning or per-target training. The model interleaves reasoning with calls to the computer algebra system SageMath: it analyzes the target graph, proposes and tests candidate constructions, and revises them until the output matches the target. The agent communicates with SageMath through a Model Context Protocol (MCP) server, which we release as a general-purpose bridge. Whether a construction matches the target is checked by a single exact isomorphism test, and therefore rests on the symbolic side and not on the model. We test the approach on a benchmark of 100 highly symmetric graphs, namely two-orbit graphs on up to 25 vertices; the benchmark was fixed in advance. Our agent could find verified algebraic constructions for all of them, without falling back to raw encodings. A strong template-enumeration baseline reaches only about $20\%$, and a catalog lookup could not identify any of these graphs. However, construction quality declines when symmetry is removed. As a concrete application, we identify the smallest known counterexample to the Bernhart-Kainen dispersability conjecture, a $16$-vertex graph that enumeration found as raw data. For this graph, our agent found an explicit algebraic construction.

Large language models (LLMs) increasingly excel at mathematical reasoning, but their unreliability limits their utility in mathematics research. A mitigation is using LLMs to generate formal proofs in languages like Lean. We perform the first large-scale evaluation of this method's ability to solve open problems. Our most capable agent autonomously resolved 9 of 353 open Erdős problems at the per-problem cost of a few hundred dollars, proved 44/492 OEIS conjectures, and is being deployed in combinatorics, optimization, graph theory, algebraic geometry, and quantum optics research. A basic agent alternating LLM-based generation with Lean-based verification replicated the Erdős successes but proved costlier on the hardest problems. These findings demonstrate the power of AI-aided formal proof search and shed light on the agent designs that enable it.

Large language models (LLMs) are increasingly used as self-study assistants in technical disciplines, yet their reliability as mathematical reasoning assistants remains poorly understood. We introduce GTBench, a curriculum-grounded benchmark for evaluating LLMs as mathematical research assistants in graph theory, comprising 63 problems organized into three groups of increasing difficulty: undergraduate definitions and basic properties (Group 1), algorithm tracing and structural reasoning (Group 2), and graduate-level proof construction (Group 3). Problems are sourced from verified academic materials including Diestel's Graph Theory. We evaluate five frontier models -- GPT-5, Claude Sonnet 4.6, Gemini 2.5 Flash-Lite, Llama 3.3 70B, and Mistral Large 3 -- under zero-shot and chain-of-thought prompting, using exact-match and LLM-as-judge evaluation for Groups 1 and 2, and a hybrid human expert and LLM-as-judge protocol for Group 3. Our results reveal a pronounced performance hierarchy: GPT-5 approaches ceiling on Group 1 (95.8% zero-shot) and maintains meaningful accuracy on graduate proofs (82%), while all other models degrade substantially with difficulty, with Llama achieving 0% under human evaluation on Group 3 zero-shot. Failure mode analysis shows that correct algorithm, wrong execution errors dominate Groups 1 and 2, while Group 3 additionally surfaces incomplete reasoning failures and reveals systematic disagreement between human evaluators and the automated judge, particularly on verbose or near-complete proofs (kappa = 0.48-0.83 across human pairs). GTBench provides the first curriculum-grounded evaluation framework for graph-theoretic reasoning in LLMs, with direct implications for the governance of AI tools in mathematical education and scientific research.