通过星翻转方法求解树的笛卡尔积的Turán数
The Turán number of the Cartesian product of trees via star-flip
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中文总结 AI 辅助
该研究证明了两棵非平凡树笛卡尔积的Turán数量级猜想，引入r-星翻转图类并推导其Turán数上界，还通过该框架给出Füredi定理的新证明。
中文摘要 AI 辅助
受Erdős关于退化二分图Turán数的猜想的启发，Bradač、Janzer、Sudakov和Tomon证明：对于任意非平凡树T和任意非平凡路径P，ex(n,T□P)=Θ_{T,P}(n^{3/2})，并猜想该量级对任意两棵非平凡树的笛卡尔积也成立。我们证明了他们的猜想。更一般地，对于每个整数r≥2，我们引入一类二分r-退化图，称为r-星翻转图，它们由初始树通过一系列局部顶点复制操作得到。我们证明：每个固定的r-星翻转图H都满足ex(n,H)=O_H(n^{2-1/r})。两棵树的笛卡尔积都是2-星翻转图，而星翻转类还包含不能作为此类乘积出现的图。作为进一步应用，我们的框架给出了Füredi定理的新证明：若H是固定二分图，其一个颜色类中至多有一个顶点的度数大于r，则ex(n,H)=O_H(n^{2-1/r})。关键要素是条件重采样过程，它将初始树上的树分支随机游走扩展到整个星翻转图的随机同态，同时在每个活动树上保留分支随机游走的分布。
英文摘要
Motivated by Erdős's conjecture on the Turán number of degenerate bipartite graphs, Bradač, Janzer, Sudakov and Tomon proved that $ \ex(n,T \Box P)=Θ_{T,P}(n^{3/2})$ for every nontrivial tree $T$ and every nontrivial path $P$, and conjectured that the same order of magnitude holds for the Cartesian product of any two nontrivial trees. We prove their conjecture. More generally, for every integer $r\ge2$, we introduce a class of bipartite $r$-degenerate graphs, called $r$-star-flip graphs, that are obtained from a seed tree by a sequence of local vertex-duplication operations. We prove that every fixed $r$-star-flip graph $H$ satisfies $\ex(n,H)=O_H(n^{2-1/r})$. Every Cartesian product of two trees is a $2$-star-flip graph, while the star-flip class also contains graphs that do not arise as such products. As a further application, our framework yields a new proof of Füredi's theorem: if $H$ is a fixed bipartite graph in which at most one vertex in one colour class has degree greater than $r$, then $\ex(n,H)=O_H(n^{2-1/r})$. The key ingredient is a conditional-resampling procedure that extends the tree branching random walk on the seed tree to a random homomorphism of the entire star-flip graph, while preserving the branching-random-walk distribution on every live tree.
