Erdős Problem #126
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
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#126 asks whether f(n)/log n → ∞, where f(n) is the fewest distinct primes that can support all pairwise sums of an n-element set — equivalently N(s)=exp(o(s)), N(s) being the largest set whose pair sums use ≤s primes. The window (½+o(1))s·log s ≤ N(s) ≤ 2^s has been open since Erdős–Turán. We didn't prove it and found no counterexample — but a strong partial: the conjecture reduces cleanly to ONE missing theorem (a uniform subexponential bound for binary S-unit equations over ℚ), the key extra structure is identified (the problem needs a clique of mutually compatible S-unit solutions, not the star a single binary equation gives), and the old local prime-by-prime route is provably dead. N(2)=4 exactly, plus explicit f(14)≤8 and f(21)≤12 — all verified on our side.
PARTIAL