Computational Exploration of Erdős Problem #43: Disjoint Sidon Sets and Asymptotic Bounds

Erdős Problem #43 asks, for Sidon sets A, B ⊆ [1,N] with (A−A) ∩ (B−B) = {0}, whether |A| + |B| ≤ f(N) (where f(N) is the maximum size of a Sidon set in [1,N]), and if |A| = |B|, whether \binom{|A|}{2} + \binom{|B|}{2} ≤ (1−c + o(1))\binom{f(N)}{2} for some c > 0. Through computational verification we confirm the negative resolution of the second part (Barreto, 2025) via Bose-Chowla constructions and small-N brute-force counterexamples. For the open first part, simulations show |A| + |B| ≤ (√2 + o(1))f(N), approaching but not exceeding Tao’s bound of ≈1.414 f(N), with ratios converging from 1.0 (q=13) to 1.266 (q=281). This provides strong computational evidence that the strong form of the first part is likely false asymptotically, while the sum approaches Tao’s √2 upper bound. This is a collaborative work between human author Britt B. and Grok (xAI), providing strong computational heuristic evidence for Erdős Problem #43 in additive combinatorics.