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Claim. The first question has a negative answer. There is no constant $C$ such that

$$\binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq\binom{f(N)}{2}+C$$

for all large $N$ and all Sidon sets $A,B\subseteq\{1,\ldots,N\}$ with $(A-A)\cap(B-B)=\{0\}$.

Covers. The first question, with its $O(1)$ error term. The second question, on pairs of equal size, is answered no on [[problems/additive_bases/E0043/claims/2025_12_19_barreto|Barreto's claim page]].

Argument. Tao noted in Problem 42's thread on 2025-12-05 that a positive answer to Problem 42, applied to a Sidon set $A$ of maximum size $f(N)$, gives for every $M$ and all $N$ large in terms of $M$ a Sidon set $B$ of size $M$ with $(A-A)\cap(B-B)=\{0\}$, so the left side exceeds $\binom{f(N)}{2}$ by $\binom{M}{2}$, which is unbounded. The proof of Problem 42 for every $M$, generated by GPT 5.5 Pro and posted by Sandhu on 2026-04-27, completes the disproof; it is recorded on [[problems/additive_bases/E0042/claims/2026_04_27_sandhu|Sandhu's claim page for Problem 42]], and this page carries the claimant and date of that solution, which the site credits. Barreto's write-up of 2026-04-29, which they had GPT produce (the preprint link), states the first question's answer as its Corollary 1.2, crediting Tao's observation, beside its Theorem 1.1 proving Problem 42 and its Theorem 1.3 for the second question; their comment of 2026-05-01 in Problem 42's thread claims the combined resolution of both questions. Bryan Kim's note An asymptotic form of Erdős Problem #43 for pairs of Sidon sets, dated 30 April 2026 and posted in this problem's thread, states the same consequence as its Theorem 1.3, conditional on the statement of Problem 42; the curator replied that Tao had already noted it. Alexeev's lean-proofs formalization of 2026-08-20, linked from Barreto's page, derives this part, not_erdos_43, from the repository's formalization of Problem 42's solution.

Acceptance. Reviewed: Thomas Bloom, the site's curator, labels the problem “DISPROVED (LEAN)” and states in the remarks that a negative answer to the first question follows from the solution of Problem 42, whose construction takes $\lvert A\rvert=f(N)$ with a companion $B$ whose size grows with $N$ (page last edited 2026-05-10; accessed 2026-10-07). No refereed publication is known; the corpus has built or reviewed none of this.

Depends on. [[problems/additive_bases/E0042/claims/2026_04_27_sandhu|Sandhu's claim page for Problem 42]], the solution the argument applies.