Computational Verification of Erdős Problem #42: Asymptotic Behavior of Maximal Sidon Sets and Disjoint Differences

Erdős Problem #42 conjectures that for any fixed positive integer n, and for all sufficiently large N (depending on n), every maximal Sidon set A ⊆ {1, …, N} admits a Sidon set B ⊆ {1, …, N} of size n such that (A−A) ∩ (B−B) = {0}. Through extensive computational exploration we identify counterexamples for small N (e.g. N=50, n=4), delineate phase transitions (around N≈90 for n=4), and conduct density and gap-geography analyses up to N=100,000. Verification against Singer sets (density ≈0.5) confirms the property holds in the densest cases. Gaps follow the O(√N) “square root law” across average, synthetic-dense, and algebraic-optimal constructions, unifying scales from N=183 to 10,303. A probabilistic heuristic grounded in this empirical evidence argues affirmatively for the conjecture asymptotically. This is a collaborative work between human author Britt B. and Grok (xAI), providing strong computational heuristic evidence for Erdős Problem #42 in additive combinatorics.