LLM Evolutionary Search Determines New Zarankiewicz Numbers
A recent publication on arXiv (2605.01120) reveals the first precise calculations of three Zarankiewicz numbers: Z(11,21,3,3)=116, Z(11,22,3,3)=121, and Z(12,22,3,3)=132. The Zarankiewicz number Z(m,n,s,t) represents the highest number of edges possible in a bipartite graph that lacks a complete K_{s,t} subgraph. Additionally, the study sets lower limits for 41 more Zarankiewicz numbers, with many being just one edge away from the best-known upper limits, and confirms values in four other closed cases. These results were achieved through OpenEvolve, an open-source evolutionary algorithm utilizing Large Language Models (LLMs) to enhance mathematical construction generation. This research highlights new extremal graph structures and showcases the capabilities of LLM-driven evolutionary search in combinatorial mathematics.
Key facts
- First exact values determined for three Zarankiewicz numbers: Z(11,21,3,3)=116, Z(11,22,3,3)=121, Z(12,22,3,3)=132.
- Lower bounds established for 41 more Zarankiewicz numbers.
- Several lower bounds are within one edge of the best known upper bound.
- Four additional closed cases match established values.
- Results obtained using OpenEvolve, an open-source LLM-based evolutionary algorithm.
- OpenEvolve iteratively improves algorithms by optimizing a reward signal tailored for the problem.
- Study provides new extremal graph constructions.
- Demonstrates potential of LLM-guided evolutionary search in mathematics.
Entities
Institutions
- arXiv