Talagrand’s original-density covering conjecture
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For a family D with p-product measure close to one on a finite ground set, must sets not contained in unions of a universal fixed number of members of D form a p-small family? This is Talagrand’s original-density covering formulation.
Why it matters and what counts as progress
Why it matters: The arbitrary-family statement is stronger than related selector conclusions and tests a discrete-convexity covering principle.
What counts as progress: A universal covering count and density threshold valid for every 0<p<1 and finite ground set, or a family violating the original formulation.
How the problem developed
- 23 Sept 2026 · Proof claim [1]
OpenAI Claimed a universal integer k for the original-density discrete-convexity covering assertion, with a formalized instance using k=2⁷⁵ for arbitrary families and 0<p<1.
AI contribution
Talagrand’s discrete-convexity conjecture
For every finite ground set, p∈(0,1) and family D of p-product measure at least 1−2^(−75), OpenAI claims the sets not contained in unions of 2^75 members of D form a p-small family.
Sources
Website timeline
Publication dates and editorial changes.
OpenAI reports Talagrand’s original-density covering theorem