Erdős Problem #708
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Erdős and Surányi's 1959 paper defined a constant cₙ — how long an interval, as a multiple of max(A), guarantees n picks whose product is divisible by ∏A — proved c₂=1 and c₃=√2, and reported no bounds in general for 67 years. Rounds 1–7 pinned c₄ through c₁₀ at 2 and proved cₙ ≥ 2 for every n ≥ 4. Round 8 proves c₁₁=2. A maximum-defect Hall reduction leaves finite eleven-edge two-, three-, and four-defect kernels. The new difficulty is placing outside-edge representatives and repair multiples jointly: multi-repair obstructions collapse by short progression lemmas, while the sole singleton obstruction is removed by a gcd/lcm exchange. Rado's theorem handles K₃,₄−e, and robust support blocks survive the aligned injection's two-adic matching while avoiding the midpoint. This determines cₙ through n=11 and improves the general bound to cₙ≤2n/11+14/11. Independent hostile audits cleared the integrated proof. The famous g(n)≤2n conjecture stays open and untouched.
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