Reinforced Generation of Combinatorial Structures: Ramsey Numbers

2026-03-10

AI Summary

This work addresses the long-standing problem of improving lower bounds for Ramsey numbers in combinatorics by introducing AlphaEvolve, a unified meta-algorithmic framework that integrates large language models, reinforcement learning, and graph-theoretic search techniques to automatically generate and optimize specialized search algorithms. By overcoming the limitations of traditional handcrafted algorithm design, this approach achieves new lower bounds for several classical Ramsey numbers: R(3,13) ≥ 61, R(3,18) ≥ 100, R(4,13) ≥ 139, R(4,14) ≥ 148, and R(4,15) ≥ 159. Moreover, it successfully reproduces all known exact lower bounds for established Ramsey numbers, thereby substantially advancing the state of the art in extremal graph theory.

Technology Category

• Search and Optimization: Metareasoning and Metaheuristics
• Machine Learning: Evolutionary Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We present improved lower bounds for five classical Ramsey numbers: $\mathbf{R}(3, 13)$ is increased from $60$ to $61$, $\mathbf{R}(3, 18)$ from $99$ to $100$, $\mathbf{R}(4, 13)$ from $138$ to $139$, $\mathbf{R}(4, 14)$ from $147$ to $148$, and $\mathbf{R}(4, 15)$ from $158$ to $159$. These results were achieved using~\emph{AlphaEvolve}, an LLM-based code mutation agent. Beyond these new results, we successfully recovered lower bounds for all Ramsey numbers known to be exact, and matched the best known lower bounds across many other cases. These include bounds for which previous work does not detail the algorithms used. Virtually all known Ramsey lower bounds are derived computationally, with bespoke search algorithms each delivering a handful of results. AlphaEvolve is a single meta-algorithm yielding search algorithms for all of our results.

Problem

• Ramsey numbers
• combinatorial structures
• lower bounds
• computational search

Innovation

• AlphaEvolve
• Ramsey numbers
• LLM-based code mutation
• combinatorial search
• automated algorithm discovery