Encoding orders and trees in real-valued functions

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.