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

TL;DR

This paper demonstrates how combining SAT solving, large language models, and formal verification can discover and prove properties of complex graphs, advancing experimental mathematics.

Contribution

It introduces a novel method integrating automated reasoning and LLMs to find and verify infinite families of doubly saturated Ramsey-good graphs.

Findings

Discovered infinite families of doubly saturated Ramsey-good graphs.

Used LLMs to generate formal correctness proofs in Lean.

Highlighted the potential of tool-driven workflows in mathematical discovery.

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.