发表机构
Department of Mathematics, Statistics, and Computer Science University of Illinois Chicago（数学、统计与计算机科学系伊利诺伊大学芝加哥分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究在实值函数中编码序与树，通过证明函数理论类似物提取阈值，第一个主要结果给出更好界并解决开放问题，第二个主要结果改进提取‘紧阈值’结果的证明及界。
AI 中文摘要
我们证明了霍奇斯关于从二元关系中编码的足够大的二叉树提取序性质的定量结果的函数理论类似物。此前，达斯卡拉克基斯和戈洛维奇以及安德森和贝内迪克特也得到了函数的类似物。这些结果来自统计学习理论，其中二叉树由顺序胖碎维度捕获，序性质由各种“阈值”概念控制。我们的第一个主要结果（定理1.11）专注于从树中提取限制较少的一种阈值，与早期关注更严格版本的结果相比，给出了显著更好的界。定理1.11的部分动机来自一篇配套论文，该定理用于获得“稳定函数”的定量正则性引理的有效界。这里将用定理1.11以更强的形式和改进的界重新证明安德森和贝内迪克特的一个结果。我们还用定理1.11证明了对偶顺序胖碎的至多双指数界，解决了一个开放问题。在我们的第二个主要结果（定理1.14）中，我们给出了达斯卡拉克基斯和戈洛维奇关于从大顺序胖碎维度提取“紧阈值”结果的新证明，界有所改进。这解决了另一个与纠正荣、金和特瓦里声称结果的证明相关的开放问题。
英文摘要
We prove function-theoretic analogues of a quantitative result of Hodges on extracting the order property from a sufficiently large 2-tree coded in a binary relation. Similar analogues for functions were previously obtained by Daskalakis and Golowich and by Anderson and Benedikt. These results are from statistical learning theory, where 2-trees are captured by sequential fat-shattering dimension, and the order property is controlled by various notions of "thresholds". Our first main result (Theorem 1.11) focuses on extracting a less restrictive kind of threshold from a tree, and yields significantly better bounds compared to what can be obtained from earlier results focusing on more restrictive versions. Part of the motivation for Theorem 1.11 lies in a companion paper, where this theorem is used to obtain efficient bounds in quantitative regularity lemmas for "stable functions". Here will use Theorem 1.11 to reprove a result of Anderson and Benedikt in a stronger form and with improved bounds. We also use Theorem 1.11 to prove an at most double-exponential bound on dual sequential fat-shattering, which resolves an open problem. In our second main result (Theorem 1.14), we give a new proof of a result of Daskalakis and Golowich on extracting "tight thresholds" from large sequential fat-shattering dimension, with improved bounds. This resolves another open problem related to correcting the proof of a result claimed by Jung, Kim, and Tewari.
Comments27 pages
