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Statement

The 3-graph $S(6)$ (p. 323, Section 1, with Fig. 1 on p. 324). On the points $1,\ldots,6$,

$$S(6)=\{123,124,345,346,561,562,135,146,236,245\},$$

ten triples, and the paper states that any four points span two triples of $S(6)$ (p. 323). In the proof of Theorem 1 (p. 325) the paper adds that the automorphism group of $S(6)$ is $A_5$ and is doubly transitive.

Example 1 (p. 323). Let $\lvert V\rvert=n$ and let $V=V_1\cup\dots\cup V_6$ be a partition. $H_S=(V,\mathcal E)$ has as edges the triples $v_{i_1}v_{i_2}v_{i_3}$ with $1\le i_1<i_2<i_3\le6$, $v_{i_j}\in V_{i_j}$ and $i_1i_2i_3\in S(6)$: one point from each of three distinct classes whose indices form a triple of $S(6)$. The paper notes (p. 324) that in $H_S$ any four points span either zero or two edges.

Edge count (p. 324). For a partition with every $\lvert V_i\rvert\ge\lfloor n/6\rfloor$, the paper states that $H_S$ has more than $10\lfloor n/6\rfloor^3$ edges, "which is more than $n^3/24$, disproving Turàn's conjecture" that $m(n,3,4,3)$ is asymptotic to $n^3/24$ (p. 323). Here $m(n,3,4,3)$ is the largest number of triples on $n$ points in which any four points span fewer than three triples, and $H_S$ qualifies because its four-point sets span zero or two.

Source. P. Frankl and Z. Füredi, An exact result for 3-graphs, Discrete Math. 50 (1984), 323--328, doi:10.1016/0012-365X(84)90058-X; $S(6)$ and Example 1 on p. 323, Fig. 1 and the edge count on p. 324. The edition is identified on the source card.

Read depth. Claims checked: the list of triples, the definition of $H_S$ and the sentences on its four-point property and edge count were read clause by clause on the page images. The four-point property of $S(6)$, which the paper leaves to the reader, was not checked here.

Proof pointer

The paper gives no proof of either four-point property; it says of $S(6)$ that "One can check" it (p. 323). The edge count is the ten triples of $S(6)$ times the product of three class sizes, each at least $\lfloor n/6\rfloor$.

Dependencies

None.

Bears on

• Problem 794: $H_S$ is the base of the iterated construction behind the lower bound of Theorem 3, which the problem page cites for the site's density reading.