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Claim. Sothanaphan's note, A compact block construction for parts 1 and 2 of Erdős Problem 12 (dated 8 April 2026, linked in the thread on 7 April, and produced with GPT-5.4 Thinking, as its disclosure states), builds $A$ from blocks $B_i=\{n:S_i\le n\le\lambda S_i,\ n\equiv r_i\pmod{M_i}\}$ with $1<\lambda<3/2$, $S_{i+1}>\lambda S_i$, distinct odd primes $q_i$, $M_i=q_1\cdots q_i$, $r_i\equiv0\pmod{q_i}$ and $r_i\equiv1\pmod{q_j}$ for $j<i$. The narrow interval rules out $a\mid b+c$ inside a block, and the tags rule it out across blocks. Theorem 5.1 gives such a set with $\liminf|A\cap\{1,\ldots,N\}|/N^{1/2}>0$; Theorem 5.2 gives one set with $|A\cap\{1,\ldots,N\}|\ge N^{1-\varepsilon}$ for every $\varepsilon>0$ and all large $N$. So the note answers the first question of Problem 12 yes and the second no. It says it simplifies DeepMind's proofs along the lines Tao suggested, and that its sets also satisfy the reading in which the two larger elements may coincide (Remark 1.2).

Submission note. Posted to the site's forum by Nat Sothanaphan on 7 April 2026:

GPT-5.4 Thinking and I have simplified DeepMind's proofs according to Tao's suggestions. We use a common construction template, which is close to Erdos-Sarkozy, for both parts.

Here are the notes.

Covers. The first two questions; not the third.

Standing. Claimed: posted in the thread and not refereed. The site's label is OPEN; its commentary credits DeepMind's construction and thanks the author.

Depends on. No page of this wiki.