Reinforced Generation of Combinatorial Structures: Ramsey Numbers

TL;DR

This paper improves lower bounds for nine classical Ramsey numbers using AlphaEvolve, an LLM-based code mutation agent, unifying and enhancing computational approaches for these combinatorial problems.

Contribution

Introduces AlphaEvolve, a meta-algorithm leveraging LLMs to generate search algorithms, leading to improved bounds for multiple Ramsey numbers and recovering known bounds.

Findings

Lower bounds for nine Ramsey numbers increased.

AlphaEvolve successfully recovered all known exact bounds.

Unified approach outperforms previous bespoke algorithms.

Abstract

We present improved lower bounds for nine 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 $148$, $\mathbf{R}(4, 15)$ from $158$ to $159$, $\mathbf{R}(4, 16)$ from $170$ to $174$, $\mathbf{R}(4, 18)$ from $205$ to $209$, $\mathbf{R}(4, 19)$ from $213$ to $219$, and $\mathbf{R}(4, 20)$ from $234$ to $237$. These results were achieved using 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.