发表机构
University of Sydney（悉尼大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文发展了乘积树的拉姆齐理论，证明了在足够大的树乘积中，大叶子子集的祖先闭包必含特定结构的算术分支副本，并推广到多叉树乘积，给出临界指数，几何上对应高维集合中的分支模式。
AI 中文摘要
我们为树的有限乘积的叶子生成子集发展了一套拉姆齐理论。我们的出发点是 Furstenberg 和 Weiss 的一个定理，该定理指出：对于每个 $k\geq 1$ 和 $\alpha>0$，如果 $A$ 是 $T_n$（高度为 $n$ 的完全二叉树）的叶子集合的子集，且大小 $|A|\geq 2^{\alpha n}$，那么当 $n$ 足够大时，由 $A$ 生成的祖先闭子树 $T_A\subset T_n$ 必定包含一个 $T_k$ 的副本，使得 (1) $T_k$ 同一层中的所有顶点被映射到 $T_A$ 同一层的顶点；(2) 如果 $T_k$ 中的非叶顶点 $x\in T_k$ 被映射到 $y \in T_A$ 中的顶点，那么 $x$ 的两个子节点被映射到 $y$ 的两个子节点的后代；(3) $T_A$ 中被 $T_k$ 副本占据的层构成一个等差数列。对于两个二叉树的乘积 $T_n \times T_n$，我们证明：对于每个 $k\geq 1$ 和 $\alpha > 1$，当 $n$ 足够大时，$T_n\times T_n$ 的叶子集合的每个大小至少为 $2^{\alpha n}$ 的子集 $A$，其祖先闭包 $\Gamma_A\subset T_n \times T_n$ 包含高度为 $k$ 的四叉树的类似结构的等差数列副本，其中源树中每个非叶顶点的四个子节点被要求映射到像顶点的四个不同的\textit{对角子节点}下方，其中顶点 $(x,y) \in T_n \times T_n$ 的对角子节点是通过在每个分量中向下移动一层得到的。我们的结果推广到 $d$ 个有限 $b$ 叉树的乘积，具有尖锐的临界指数 \\[ \alpha_{\text{crit}}(d,b) =d-1 + \log_b(b-1).\\] 从几何上看，我们的结果意味着：任何上 Minkowski 维数大于 $\alpha_{\text{crit}}(d,b)$ 的集合 $E\subset [0,1)^d$ 包含任意有限阶的算术 $b$ 进分支模式。
英文摘要
We develop a Ramsey theory for leaf-generated subsets of finite products of trees. Our starting point is a Theorem of Furstenberg and Weiss which states that for every $k\geq 1$ and $α>0$, if $A$ is a subset of the leaves of $T_n$, where $T_n$ is the complete binary tree of height $n$, with size $|A|\geq 2^{αn}$, then for $n$ sufficiently large the ancestor closed sub-tree $T_A\subset T_n$ generated by $A$ must contain a copy of $T_k$ such that (1) all vertices in the same level of $T_k$ are mapped into vertices at the same level of $T_A$, (2) if a non-leaf vertex $x\in T_k$ is mapped into a vertex $y \in T_A$, then the two children of $x$ are mapped into descendants of the two children of $y$, and (3) the levels of $T_A$ occupied by the copy of $T_k$ form an arithmetic progression. For the product $T_n \times T_n$ of two binary trees, we show that for every $k\geq 1$ and $α> 1$, for $n$ sufficiently large, every subset $A$ of the leaves of $T_n\times T_n$ of size at least $2^{αn}$ has that its ancestor closure $Γ_A\subset T_n \times T_n$ contains similarly structured arithmetic copies of the height $k$ four-ary tree, where the four children of every non-leaf vertex in the source tree are required to map below the four distinct \textit{diagonal children} of the image vertex, where a diagonal child of a vertex $(x,y) \in T_n \times T_n$ is obtained by moving one level down each in each component. Our results generalise to product of $d$-many finite $b$-ary trees with a sharp critical exponent of \[ α_{\text{crit}}(d,b) =d-1 + \log_b(b-1).\] Geometrically, our results imply that any set $E\subset [0,1)^d$ of upper Minkowski dimension greater than $α_{\text{crit}}(d,b)$ contains arithmetic $b$-adic branching patterns of arbitrary finite order.
