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Statement

Construction (Section 5, p. 21; unnumbered). Let $c=\lceil n/2\rceil$ and

$$\mathcal C_1=\{S\subset[1,n]:\ |S|\le3\text{ and }c\in S\},$$

a well-intersecting family (Definition 1, p. 3: every two distinct members meet in a non-empty arithmetic progression) with $|\mathcal C_1|=\binom n2+1$. For each integer $x$ with $1\le x\le\left[\frac{n-1}{4}\right]$, add to $\mathcal C_1$ the three sets

$$\{c-2x,c-x,c,c+x,c+2x\},\qquad\{c-2x,c-x,c,c+x\},\qquad\{c-x,c,c+x,c+2x\},$$

and remove the two triples $\{c-2x,c,c+x\}$ and $\{c-x,c,c+2x\}$. The resulting family $\mathcal C$ is well-intersecting, and the paper computes

$$|\mathcal C|=|\mathcal C_1|+\left[\frac{n-1}{4}\right] =\binom n2+\left[\frac{n-1}{4}\right]+1,$$

where $[\cdot]$ is the paper's integer-part bracket. Hence

$$N_1(n)\ge\binom n2+\left[\frac{n-1}{4}\right]+1,$$

which exceeds $\binom n2+1$ for $n\ge5$. The paper states (p. 21) that it had been conjectured that $\mathcal C_1$ is an extremal family for $N_1$; the construction disproves that conjecture (abstract, p. 1, and p. 3).

A further family (p. 21). For $X\subseteq\left[1,\left[\frac{n-1}{8}\right]\right]$ the paper builds a well-intersecting family $\mathcal C_X$ by adding to $\mathcal C$, for every $x\in X$, the nine-term progression $\{c-4x,c-3x,\ldots,c+4x\}$ with all of its subprogressions containing $c$ (16 new sets for each $x$) and leaving out the 16 triples that would violate the well-intersecting property, and states $|\mathcal C_X|=|\mathcal C|$. It gives this as one of several equally good constructions, one that contains an arithmetic progression of length nine.

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 5 on p. 21.

Read depth. Claims checked: the construction of $\mathcal C$ and its count were read clause by clause on the page image. The paper gives no written check that $\mathcal C$ is well-intersecting; as an observation of this page, not of the paper, an exhaustive pairwise check of $\mathcal C$ for $3\le n\le25$ found every pairwise intersection a non-empty arithmetic progression and $|\mathcal C|=\binom n2+\left[\frac{n-1}{4}\right]+1$. The family $\mathcal C_X$ was read but not checked. Nothing here is independently reviewed.

Proof pointer

Page 21. For each $x$ three sets enter and two leave, so each admissible $x$ adds one set; the sets entering for different $x$ have different common differences and are distinct. The two removed triples are exactly those of $\mathcal C_1$ that meet one of the added four- or five-term sets in a non-progression.

Dependencies

None beyond Definition 1 of the same paper.

Bears on

• Problem 272: with the sets read as distinct, a lower bound $\binom N2+\left[\frac{N-1}{4}\right]+1$ on the problem's largest $t$, which shows that the family of all sets of at most three elements through a fixed point does not attain it for $N\ge5$. It does not give the exact value.