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Statement

Theorem 2 (p. 324, quoted). "Suppose that $H=(V,\mathcal E)$, $\lvert V\rvert=n\ge5$ and any 4 points of $V$ span 0 or 2 edges. Then $\max\lvert\mathcal E\rvert$ is attained exactly for $H$ of the form $H_S$ for some equipartition, i.e., $\lfloor n/6\rfloor\le\lvert V_i\rvert\le\lceil n/6\rceil$."

Here $H_S$ is the blow-up of $S(6)$ over a partition $V=V_1\cup\dots\cup V_6$ (Example 1, p. 323), and a hypergraph has every vertex in some edge (p. 323). The argument on p. 327 adds that the Example 2 bound equals the Example 1 maximum only for $n\le5$, where "the two examples coincide"; so at $n=5$ the extremal $H_S$ is also of the circle form of Example 2.

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; Theorem 2 on p. 324, its deduction from Theorem 1 on p. 327. The edition is identified on the source card.

Read depth. Claims checked: the statement and the p. 327 comparison were read clause by clause on the page images. The count behind the comparison was not replayed, and the paper writes out no argument that the equipartition maximizes the edge count of $H_S$.

Proof pointer

P. 327. By Theorem 1 an extremal 3-graph is of the form of Example 1 or Example 2. For Example 2, with $d_1,\ldots,d_{2k+1}$ points placed on the vertices of a regular $(2k+1)$-gon, the edge count is largest when the $d_i$ are as equal as possible, which bounds it by $(n/(2k+1))^3(2k+1)\binom{k+1}2/3$, less than $n^3/24$; the paper states that this never exceeds the largest 3-graph of Example 1, with equality only for $n\le5$.

Dependencies

Theorem 1; Example 1.

Bears on

• Problem 794: a 3-graph whose four-point sets span zero or two edges has no four vertices spanning three edges, so it counts toward $m(n,3,4,3)$, the quantity of the site commentary's reading. Theorem 2 shows that within this class nothing beats $H_S$; the larger iterated construction behind the lower bound of Theorem 3 leaves the class: a triple added inside $V_1$ together with a vertex of $V_2$ spans exactly one edge (an observation of this page). The problem page names Theorems 1--2 only as context, recording that they show $H_S$ over an equipartition extremal for $n\ge5$ under the stricter local condition.