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Comparison of fractional and integral expectation thresholds

Are fractional and integral expectation thresholds universally comparable for every nonempty proper increasing family on a finite nonempty ground set? Both definitions use the same covering budget one half, and fractional support is unrestricted.

Why it matters and what counts as progress

Why it matters: A universal constant comparison relates fractional covering relaxations to integral expectation thresholds without structural restrictions on the family.

What counts as progress: Prove q_f(F)≤Cq(F) for an absolute C, improve the constant or identify when the thresholds coincide.

How the problem developed

  1. 2010 · Conjecture [1]

    Michel Talagrand Proposed that the integral and fractional expectation thresholds of every increasing family are comparable within a universal constant.

  2. 13 Aug 2021 · Partial bound [1]

    Keith Frankston, Jeff Kahn, Bhargav Narayanan and Jinyoung Park Proved a logarithmic-factor bound relating the random threshold to the fractional expectation threshold, extending several results in probabilistic combinatorics.

  3. 23 Sept 2026 · Reported constant-factor theorem [2]

    OpenAI Reported a universal constant comparison between the integral and fractional expectation thresholds for every nonempty proper increasing family on a finite set.

AI contribution

Sources

  1. Frankston, Kahn, Narayanan and Park · Thresholds versus fractional expectation-thresholds (2021)
  2. OpenAI, Talagrand's conjecture that fractional and integral expectation thresholds agree up to a constant
  3. Integral and fractional expectation thresholds are equivalent
  4. OpenAI mathematics release and provenance
  5. Lean proof of OpenAI reports a constant-factor expectation-threshold theorem
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