New Bounds for Zarankiewicz Numbers via Reinforced LLM Evolutionary Search

TL;DR

This paper introduces new exact and lower bound results for Zarankiewicz numbers using an open-source LLM-guided evolutionary algorithm, demonstrating a cost-effective approach to combinatorial research.

Contribution

It presents the first exact values for three Zarankiewicz numbers and establishes lower bounds for 41 others using a novel LLM-based evolutionary search method.

Findings

Exact values for three Zarankiewicz numbers determined.

Lower bounds established for 41 additional numbers.

Cost of computation less than $30 per parameter set.

Abstract

The Zarankiewicz number $\textbf{Z}(m, n, s, t)$ is the maximum number of edges in a bipartite graph $G_{m, n}$ such that there is no complete $K_{s, t}$ bipartite subgraph. We determine for the first time the exact values of three Zarankiewicz numbers: $\textbf{Z}(11, 21, 3, 3)=116$, $\textbf{Z}(11, 22, 3, 3)=121$, and $\textbf{Z}(12, 22, 3, 3)=132$. We further establish lower bounds for 41 more Zarankiewicz numbers, including several that are within one edge of the best known upper bound, and we match the established value in four more closed cases. Our results are obtained using OpenEvolve, an open-source evolutionary algorithm based on Large Language Models (LLMs) that iteratively improves algorithms for generating mathematical constructions by optimizing a reward signal which we tailored for this specific problem. These findings provide new extremal graph constructions and demonstrate the potential of LLM-guided evolutionary search to contribute to mathematical research. In addition to presenting the resulting constructions, we report the generation algorithms produced, describe the relevant implementation details, and provide our computational costs. Our costs are remarkably low, at less than \$30 for each Zarankiewicz parameter combination, showing that LLM-guided evolutionary search can be an inexpensive, reproducible, and accessible tool for discovering new combinatorial constructions.