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Claim. Let $t(N)$ be the largest number of subsets of $\{1,\ldots,N\}$ whose pairwise intersections are all nonempty arithmetic progressions, the quantity Problem 272 asks for. Theorem 1.1 of Zhanfu Yang, Exact values and exact upper bounds for families of integers with arithmetic progression intersections (Erdős Problem #272), arXiv:2607.23004 (version 1 of 2026-07-25; card Yang 2026), states

$$t(N)=4,\ 7,\ 12,\ 17,\ 23,\ 30,\ 39,\ 48,\ 58,\ 69 \qquad(N=3,4,\ldots,12),$$

so that Szabó's lower bound $\binom N2+1+\lfloor(N-1)/4\rfloor$ is exact in that range. The statement is computer-assisted: Section 6 describes maximum-clique computations for $N\le10$ and decision searches excluding cliques of sizes $59$ and $70$ for $N=11$ and $12$. The values for $N\le9$ had been posted on the site's discussion thread in August 2025, as the paper itself says; the new values are $t(10)=48$, $t(11)=58$ and $t(12)=69$. Theorem 1.4 proves that every family whose members share a common element (a starred family) has at most $\binom N2+1+\lfloor(N-1)/4\rfloor$ members, so by Szabó's construction that is the exact maximum over starred families for every $N\ge1$; Corollary 5.4 reduces the paper's Conjecture 1.3, that this formula is $t(N)$ for every $N$, to the existence form of Szabó's kernel question. The author discloses the use of Claude for some computations and drafting, and states that the author checked all proofs and computational claims.

Covers. The exact value of $t(N)$ for $3\le N\le12$, and the exact maximum over families with a common element for every $N$. Not covered: $t(N)$ for $N\ge13$, the kernel question, and Conjecture 1.3.

Depends on. Nothing in this wiki: the lower bound used is Szabó's construction, which the paper presents in full in its Section 2.

Standing. Claimed. The preprint is unrefereed; the computations behind Theorem 1.1, whose code repository the paper names in Section 6, have not been inspected or rerun by this corpus; the site labels the problem OPEN and its commentary does not mention the paper; and no outside review is recorded, so no evidence kind is listed.