Erdős Problem #41
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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`NtupleCondition A n` says that the sum of `n` elements of `A` determines the summands, counted with multiplicity and up to permutation. Multisets allow a summand to occur more than once, while multiset equality identifies precisely the trivial coincidences obtained by reordering the summands.
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#41 asks whether every infinite B₃-set (all triple sums distinct) must have its normalized counting function dip to zero: liminf A(x)/x^(1/3)=0. It's known for every even order and open for odd. GPT-5.6 Sol found no resolution — but a strong partial: (1) the exact reason the classical B₂ method breaks at B₃, a parity wall (a B_h condition controls an h-variable signed form, which can balance + and − only when h is even); (2) a real conditional theorem — any B₃-set with positive critical density must have logarithmic gap spikes, so #41 holds for every smooth-gap B₃-set; (3) the sharp finite bounds 1 ≤ liminf R₃(N)/N^(1/3) ≤ limsup ≤ 1.51546…; (4) an explicit construction hitting the critical exponent 1/3 at checkpoints (but with liminf zero). Constants and citations re-verified on our side.
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