P-smallness of Increasing Families
summary
Defines p-smallness as an explicit certificate: an increasing family F is p-small when a cover G satisfies Σ p^{|B|} ≤ 1/2. Presents a theorem giving a universal K with pc(F) ≤ K·q_F·log l(F) (threshold ≤ K·Expectation Threshold·log largest minimal size). Uses spread/fractional relaxations and sunflower-inspired tools to remove a log factor in axial d‑dimensional random assignment (e.g. Z3(n) = Θ(1/n) rather than Θ((log n)/n)). Also proves a weighted sup deviation event is fractionally small.
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