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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 sea","title":"New Bounds for Zarankiewicz Numbers via Reinforced LLM Evolutionary Search","url":"https://arxiv.org/abs/2605.01120","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.01120v2 Announce Type: replace \nAbstract: 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. 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