AI 中文总结
该研究针对二部图类限制的树极值数问题，通过构造下界、提出上界猜想，利用k-极小性加权变体证明了多类树的上界，关联了固定二分划与普通极值数，还探讨了类似Zarankiewicz函数的有向二部极值函数。
AI 中文摘要
我们研究经典Erdős–Sós问题的一个限制情形，即树的极值数，该限制针对二部宿主图类，分别在仅规定宿主图阶数、以及固定其二分划两部分大小两种情形下展开。我们给出自然的下界构造，并提出相应的线性上界猜想。我们应用k-极小性的加权变体，证明了一大类树的上界，包括扫帚树、二分划两部分大小差不超过1的树，以及所有顶点数不超过7的树，在加性常数的误差范围内解决了Caro、Patkós和Tuza提出的部分问题。我们还将树的固定二分划极值数与普通极值数关联起来，并考虑了类似Zarankiewicz函数的有向二部极值函数。
英文摘要
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.
Comments13 pages, 1 figure
