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Dual Sequential Fat-Shattering and Tight Threshold Extraction

Two open problems about extracting order from trees in real-valued functions. A quantitative function analogue of Hodges's tree-to-order extraction yields an at most double-exponential bound on dual sequential fat-shattering dimension, resolving the first. A new proof of Daskalakis-Golowich tight-threshold extraction, avoiding multicolored Ramsey numbers, resolves the second, which concerned repairing the bound in a result claimed by Jung, Kim and Tewari.

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theoretical-computer-scienceJul 23, 2026Significance 10/100Registry: unreviewed

Dual Sequential Fat-Shattering and Tight Threshold Extraction

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Two open problems about extracting order from trees in real-valued functions. A quantitative function analogue of Hodges's tree-to-order extraction yields an at most double-exponential bound on dual sequential fat-shattering dimension, resolving the first. A new proof of Daskalakis-Golowich tight-threshold extraction, avoiding multicolored Ramsey numbers, resolves the second, which concerned repairing the bound in…

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Two open problems about extracting order from trees in real-valued functions. A quantitative function analogue of Hodges's tree-to-order extraction yields an at most double-exponential bound on dual sequential fat-shattering dimension, resolving the first. A new proof of Daskalakis-Golowich tight-threshold extraction, avoiding multicolored Ramsey numbers, resolves the second, which concerned repairing the bound in a result claimed by Jung, Kim and Tewari.

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