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Statement

Questions (Section 6, pp. 21--22; unnumbered). Section 6 opens (p. 21) by naming the exact value of $N_1$, ideally with a description of the extremal systems, as the goal. Since the families of Section 5 arise from $\mathcal C_1$ by adding and removing $O(n)$ members, the paper suggests that extremal systems differ only slightly from a family of the type of $\mathcal C_1$ (all sets of at most three elements through a fixed integer), and asks two precise questions (p. 22, quoted): "can one prove that every element of an extremal system contains a fixed integer $c$? Is it true that $N_1\le n^2/2+O(n)$?"

Here $N_1=N_1(n)$ is the largest size of a family of subsets of $[1,n]$ in which every two distinct members meet in a non-empty arithmetic progression (Definition 1, p. 3). Since the Section 5 lower bound $\binom n2+\left[\frac{n-1}{4}\right]+1$ is $n^2/2+O(n)$, an affirmative answer to the second question would give $N_1=n^2/2+O(n)$; this deduction is this page's, not the paper's.

The same section (p. 22) also records as still open, to the author's knowledge, a question of Simonovits and Sós: whether the extremal systems for $N_k$, $k\ge2$, contain arithmetic progressions only.

Source. Tibor Szabó, Intersection properties of subsets of integers, European J. Combin. 20 (1999), no. 5, 429--444, DOI 10.1006/eujc.1997.0176. Pages are those of the author's 23-page preprint identified on the source card, not of the journal edition: Section 6 on pp. 21--22.

Read depth. Claims checked: Section 6 was read in full on the page images. Nothing here is independently reviewed.

Dependencies

Theorem 2.1 and the Section 5 construction of the same paper frame the questions.

Bears on

• Problem 272: both questions concern the problem's extremal families, read with distinct sets. An affirmative answer to the second would determine the problem's largest $t$ up to $O(N)$, not exactly; the first asks about the structure of the extremal families. The problem page records the standing of each.