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

2026-04-22

AI Summary

This study resolves an open problem posed by Grinstead and Roberts in 1982 concerning the existence of Ramsey-good graphs that are doubly saturated—namely, graphs containing neither a clique of size $s$ nor an independent set of size $t$, such that adding or removing any edge destroys this property. We introduce a novel, fully automated pipeline that deeply integrates SAT solving, large language models (LLMs), and formal verification in Lean, spanning conjecture generation, graph construction, and machine-checkable proof production. This approach systematically uncovers infinite families of doubly saturated Ramsey-good graphs and formally verifies their correctness within the Lean proof assistant, thereby advancing a new paradigm of AI-driven experimental mathematics.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

Ramsey-good graphs are graphs that contain neither a clique of size $s$ nor an independent set of size $t$. We study doubly saturated Ramsey-good graphs, defined as Ramsey-good graphs in which the addition or removal of any edge necessarily creates an $s$-clique or a $t$-independent set. We present a method combining SAT solving with bespoke LLM-generated code to discover infinite families of such graphs, answering a question of Grinstead and Roberts from 1982. In addition, we use LLMs to generate and formalize correctness proofs in Lean. This case study highlights the potential of integrating automated reasoning, large language models, and formal verification to accelerate mathematical discovery. We argue that such tool-driven workflows will play an increasingly central role in experimental mathematics.

Problem

• Ramsey-good graphs
• doubly saturated
• clique
• independent set
• mathematical discovery

Innovation

• SAT solving
• large language models
• formal verification
• Ramsey graphs
• computer-assisted discovery