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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 2.1 of Szabó, recorded on the card Szabó 1999, gives $t(N)<N^2/2+O(N^{5/3}\log^3N)$, which with the lower bound of the paper's own Section 5 construction below, at least the $\binom N2+1$ members of the family of Simonovits and Sós, yields

$$t(N)=\frac{N^2}{2}+O\!\left(N^{5/3}(\log N)^3\right),$$

the leading asymptotic $t(N)=(1/2+o(1))N^2$, which the site's commentary calls the resolution of the asymptotic question. The proof splits a family into progressions and non-progressions, bounds the large non-progressions (Lemma 2.2: determining triples counted through Lemma 1 of Simonovits and Sós, and a separate count for a progression plus one point) and, by Theorem 4 of Simonovits and Sós, the small non-progressions when they have empty common intersection, and, when the small non-progressions share a point $c$, attaches to each a determining triple through $c$ whose two essential elements are counted like the endpoints of a progression; a gcd count over pairs of differences couples the two types and the endpoint sums give the $N^2/2$ main term. Section 5 of the paper gives, for $c=\lceil N/2\rceil$, a family obtained from the sets of at most three elements through $c$ by adding, for each $1\le x\le\lfloor(N-1)/4\rfloor$, the five-term progression $\{c-2x,c-x,c,c+x,c+2x\}$ and its two four-term subprogressions through $c$ and deleting the two triples through $c$ that obstruct them; it has

$$\binom N2+\left\lfloor\frac{N-1}{4}\right\rfloor+1$$

members, which refutes the conjecture of Simonovits and Sós that $\binom N2+1$ is the exact value. Section 6 asks two questions: whether every member of an extremal family contains a fixed integer (the kernel question) and whether $t(N)\le N^2/2+O(N)$ (the linear-error question, answered yes on JenW1N's claim page).

Covers. The asymptotic $t(N)=N^2/2+O(N^{5/3}(\log N)^3)$, the lower bound $t(N)\ge\binom N2+\lfloor(N-1)/4\rfloor+1$, and the refutation of the Simonovits–Sós conjecture. Not covered: the exact value of $t(N)$, which the catalog question asks for, the linear error term and the kernel question.

Depends on. Nothing in this wiki: the proof uses Lemma 1 and Theorem 4 of Simonovits and Sós as cited inputs, and the earlier claim page records only that paper's bounds, not an input the asymptotic rests on.

Acceptance. Refereed: European Journal of Combinatorics 20 (1999), no. 5, 429--444, the DOI linked above; the publication record dates the issue to July 1999, filled to the first of the month for this page's name. Reviewed is not listed: the site labels the problem OPEN, and its commentary crediting Szabó with the asymptotic and the refutation is commentary on an open problem, not acceptance of a solution. Formalized is not listed: the formal-conjectures catalog states the asymptotic as the variant szabo of its file for the problem and marks it research solved, but states it without a proof, and this corpus has built no proof of it.