Bipartite Extremal Numbers of Trees

Lucas Waite; Nuh Aydin

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arXiv · 2607.29579

Bipartite Extremal Numbers of Trees

Lucas Waite·Nuh Aydin

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Abstract

We study a restriction of the classical Erdős--Sós problem, the extremal number of trees, to the class of bipartite host graphs, both when only the order of the host is prescribed and when its two part-sizes are fixed. We give natural lower-bound constructions and formulate corresponding linear upper-bound conjectures. We apply a weighted variant of $k$-minimality to prove upper bounds for a broad family of trees including brooms, trees with part-sizes obeying certain inequalities, and all trees on at most seven vertices, resolving part of a problem of Caro, Patkós and Tuza up to additive constants. We also relate the fixed-part extremal number of a tree to the ordinary extremal number, and consider an oriented bipartite extremal function analogous to the Zarankiewicz function.

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Lucas Waite, Nuh Aydin. 2026-08-10. Bipartite Extremal Numbers of Trees. https://arxiv.org/abs/2607.29579

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