# Reinforced Generation of Combinatorial Structures: Ramsey Numbers

arXiv: https://arxiv.org/abs/2603.09172 · HTML: https://mlsanity.com/paper/2603.09172/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 15 references that are in the map.
- First posted on arXiv 10 Mar 2026.

## Builds on (most cited first, 6 of 6)

- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)
- [Mathematical exploration and discovery at scale](https://mlsanity.com/paper/2511.02864/) (2025)
- [Constructions in combinatorics via neural networks](https://mlsanity.com/paper/2104.14516/) (2021)
- [PatternBoost: Constructions in Mathematics with a Little Help from AI](https://mlsanity.com/paper/2411.00566/) (2024)
- [Reinforced Generation of Combinatorial Structures: Hardness of Approximation](https://mlsanity.com/paper/2509.18057/) (2025)
- [A Formal Proof of R(4,5)=25](https://mlsanity.com/paper/2404.01761/) (2024)

## Built on by (most cited first, 9 of 9)

- [Advancing Mathematics Research with AI-Driven Formal Proof Search](https://mlsanity.com/paper/2605.22763/) (2026)
- [New Bounds for Zarankiewicz Numbers via Reinforced LLM Evolutionary Search](https://mlsanity.com/paper/2605.01120/) (2026)
- [ScienceIDE: Turning World's Scientific Codebase into Agent Learnable Environments](https://mlsanity.com/paper/2609.19134/) (2026)
- [pAI/MSc: ML Theory Research with Humans on the Loop](https://mlsanity.com/paper/2604.20622/) (2026)
- [k-server-bench: Automating Potential Discovery for the k-Server Conjecture](https://mlsanity.com/paper/2604.07240/) (2026)
- [Genetic Algorithms with Optimization Guided Operators](https://mlsanity.com/paper/2606.12279/) (2026)
- [Stable Menus of Public Goods: AI-Enabled Progress](https://mlsanity.com/paper/2606.16989/) (2026)
- [EvoMem: Memory-Augmented Evolution for Code Optimization](https://mlsanity.com/paper/2608.10795/) (2026)
- [RamseyGadgets: A Graph Construction Dataset for LLMs](https://mlsanity.com/paper/2608.14999/) (2026)

## Abstract (arXiv)

We present improved lower bounds for nine classical Ramsey numbers: R(3, 13) is increased from 60 to 61, R(3, 18) from 99 to 100, R(4, 13) from 138 to 139, R(4, 14) from 147 to 148, R(4, 15) from 158 to 159, R(4, 16) from 170 to 174, R(4, 18) from 205 to 209, R(4, 19) from 213 to 219, and 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.