New Bounds for Zarankiewicz Numbers via Reinforced LLM Evolutionary Search

2026-05-01

AI Summary

This work investigates the exact values and lower bounds of Zarankiewicz numbers \( Z(m,n,s,t) \), which denote the maximum number of edges in a bipartite graph avoiding a complete bipartite subgraph \( K_{s,t} \). We introduce OpenEvolve, a large language model–guided reinforcement-based evolutionary search framework that automatically constructs extremal graph configurations through tailored reward mechanisms and self-generated algorithmic strategies. Using this approach, we determine for the first time the exact values \( Z(11,21,3,3) = 116 \), \( Z(11,22,3,3) = 121 \), and \( Z(12,22,3,3) = 132 \); establish new lower bounds for 41 instances—several within a single edge of known upper bounds; and reproduce four previously reported results. All computations cost less than \$30 per run, demonstrating the efficiency and cost-effectiveness of LLM-driven evolutionary search in advancing combinatorial discovery.

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.

Problem

• Zarankiewicz numbers
• bipartite graph
• extremal graph theory
• combinatorial constructions
• K_{s,t}-free

Innovation

• Zarankiewicz numbers
• LLM-guided evolutionary search
• extremal graph constructions
• OpenEvolve
• combinatorial optimization