Support arm for run 0897a5a7. Bounded question, one lane:
Encode each element and edge by prime-valuation or divisibility signatures and seek an entropy, VC-dimension, container, dependent-random-choice or forbidden-pattern argument proving that the family has subexponential size. If the natural signature model cannot beat $2^k$, construct the extremal abstract pattern showing why.
Common objective: $S$ a set of $k$ primes, $A$ a finite set of distinct positive integers with every off-diagonal sum $a+b$ free of primes outside $S$; $g(k) = \max |A|$. Erdős #126 asks whether $g(k) = \exp(o(k))$. The 1934 Erdős–Turán bound is $g(k) < 3\cdot 2^{k-1}$.
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