#41: strong partial — the exact parity wall between B₂ and B₃, plus a real conditional theorem
The target (erdosproblems.com/41, OPEN, $500). A B₃-set is a set of integers in which all triple sums aᵢ+aⱼ+aₖ (repetition allowed) are distinct. Write A(x) for the counting function of such a set. The question: must every infinite B₃-set have its normalized density dip to zero along a subsequence — is liminf A(x)/x^(1/3) = 0? For B₂ (Sidon sets) and every even order this "logarithmic dip" is known; #41 is the odd-order case h=3, and it is open.
We did not resolve it. The status check is unusually clean, and the partial package is real. A brand-new primary source — Christian Táfula, "Infinite Sidon-type sets for zero-sum linear forms" (arXiv:2607.20753, July 2026) — recovers the full logarithmic dip for every even order through zero-sum linear forms, and states explicitly that the corresponding problem for odd h remains open. So #41 has not quietly been subsumed by a general B_h theorem; the parity is genuinely the obstacle. We confirmed the paper and its author are real, and re-derived the load-bearing constants and the proof structure below on our side.
The diagnosis: an exact parity wall, not a missing trick. The classical B₂ argument is bare-bones: if a Sidon set had liminf A(x)/√x > 0 then aₙ ≪ n²; take ≍ log x geometrically separated blocks, each of span O(N²) partitioned into O(N²/x) intervals of length x; Cauchy–Schwarz forces ≫ x equal-difference pairs per block, the Sidon property makes all those differences distinct, and log x blocks give ≫ x log x distinct integers in [1,x] — impossible. The whole engine runs on one cancellation: translate a block by t and the differences (a+t)−(b+t) = a−b don't move. For B₃ the useful injective expression is a+b−c, and translating gives (a+t)+(b+t)−(c+t) = a+b−c+t — the block location does NOT cancel, so the differences scatter instead of accumulating. The Fourier version is the same wall: the even-order integrand is a product of matched |F|² factors, nonnegative, so each dyadic block keeps its major-arc contribution and the blocks add; the B₃ form carries F(α)²·F̄(α) = F(α)|F(α)|², an uncontrolled phase with no nonnegative blockwise lower bound to sum. The clean statement of the wall: a B_h condition naturally controls a signed form in h variables, and such a form can have equally many + and − variables only when h is even.
A real conditional theorem that falls out anyway. The run proves: if a B₃-set has positive critical density, liminf A(x)/x^(1/3) > 0, then it must have logarithmically superquadratic gap spikes — limsup (aₙ₊₁−aₙ)/(n²log n) > 0, hence limsup (aₙ₊₁−aₙ)/n² = ∞. Equivalently, #41 has a positive answer for every B₃-set whose gaps are smooth (aₙ₊₁−aₙ = o(n²log n)). The mechanism: positive density pins aₙ between c·n³ and C·n³; a block-crossing count on the injective sums aᵢ+aⱼ−aₖ then forces ≫ x·min(ξ,log x) distinct values into [−x,x] if the gaps stay below n²log n/ξ with ξ→∞, contradicting |[−x,x]∩ℤ| = 2x+1. The honest label: this is a direct synthesis of Li's finite-B₃ injectivity observation and the block-crossing architecture in Táfula's gap theorem, not a novelty claim. What it buys is genuine but not fatal: any counterexample must be dense despite violent sparse discontinuities, not dense and regular.
The finite theory is healthier than the infinite one. Let R₃(N) be the largest B₃-subset of [1,N]. O'Bryant's construction gives R₃(N) ≥ N^(1/3) − N^(7/40) for large N, so liminf R₃(N)/N^(1/3) ≥ 1. The best verified upper constant combines White's B₃ bound with Rechnitzer's 2026 rigorous computation of the L²-autoconvolution constant ν₂ (arXiv:2602.07292, "The first 128 digits of an autoconvolution inequality"), which encloses ν₂² = 0.5746396071515195… to 128 decimal digits:
1 ≤ liminf R₃(N)/N^(1/3) ≤ limsup R₃(N)/N^(1/3) ≤ (2/ν₂²)^(1/3) = 1.51546116978632…
We re-checked that last arithmetic directly: (2/0.5746396071515195…)^(1/3) reproduces 1.51546116978632 to the displayed digits. The limit R₃(N)/N^(1/3) is not known to exist.
An explicit construction hits the critical exponent — but only at checkpoints. A scale-separated gluing lemma (if A⊆[0,P] and B⊆[0,L] are finite B₃-sets with 0∈B, then A ∪ (t + d·B) is B₃ for explicit d,t that make the four triple-sum ranges disjoint) lets one glue generalized Bose–Chowla blocks over 𝔽_{q³} into an infinite B₃-set with limsup log A(x)/log x = 1/3 — the critical exponent itself — and in fact A(N_k) ≥ N_k^(1/3)/(log N_k)^(o(1)) along infinitely many checkpoints, with the normalized checkpoint density decaying as slowly as prescribed. The catch, stated plainly: this construction has liminf zero. Right before each new block the counting function is still tiny while the next block starts at a vastly larger scale, so it produces enormous valleys. It does not refute #41; it lives exactly through the loophole the gap proposition exposes (gigantic inter-block gaps), and shows a proof of #41 can only force a sparse subsequence of dips, never a uniform rate. The strongest genuinely uniform infinite construction is Cilleruelo–Tesoro's (arXiv:1206.3087), A(x) = x^(√5−2+o(1)) = x^(0.2360679…), far below the critical 1/3 — we re-checked √5−2 = 0.2360679… as well.
What's still open. Everything above is a bounded search, a conditional theorem, or a status reduction; no uniform statement that every infinite B₃-set dips was proved, and the remaining case — dense despite violent sparse discontinuities — is exactly what the parity wall leaves open. The finite-field search over 𝔽₅³ and 𝔽₁₁³ Bose–Chowla families found no anomaly (a 32-element best diameter at q=5), but that is computation, not a theorem. #41 stays open; what changed is that the obstruction now has a name (the even/odd parity wall), a conditional positive result (smooth gaps ⟹ the dip), and a sharp finite window [1, 1.51546…] around the unknown limiting constant.