[Submitted on 4 Dec 2024 (v1), last revised 6 Sep 2026 (this version, v2)]

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Abstract:We prove a sharp version of Talagrand's selector process conjecture. Roughly speaking, given any collection of nonnegative weight vectors whose support form a family that is not $p$-small, a random set of density $O(sp)$ captures all but a $2^{-s}$ fraction of weight of some vector with high probability. This gives a common strengthening of Talagrand's selector process conjecture and the Kahn--Kalai conjecture.
We give two applications of this result. First, towards a conjecture of Talagrand on the equivalence of expectation thresholds and fractional expectation thresholds, we show that a $p$-small fractional cover supported on sets of size at most $t$ can be rounded to a $cp/\log(2t)$-small integral cover. As a corollary, we show that the fractional and integral expectation thresholds are separated by at most a $\log \log$ factor. Second, we prove a Sudakov minoration principle for general positive selector processes, which in particular resolves a problem of Talagrand on Sudakov minoration for the product Bernoulli measure.

Submission history

From: Huy Tuan Pham [view email]
[v1] Wed, 4 Dec 2024 18:37:17 UTC (13 KB)
[v2] Sun, 6 Sep 2026 22:06:17 UTC (23 KB)

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