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Statement

Conventions (p. 323): a hypergraph $H=(V,\mathcal E)$ has $\bigcup\mathcal E=V$, so every vertex lies in an edge, and it is a 3-graph when every edge has three elements; the edges spanned by $W\subset V$ are those contained in $W$.

Theorem 1 (p. 324, quoted). "Suppose $H=(V,\mathcal E)$ is a 3-graph in which any 4 points span 0 or 2 edges. Then $H$ is isomorphic to one of the 3-graphs in Examples 1 or 2."

The two families:

• Example 1 (p. 323): the blow-up $H_S$ of the ten-triple 3-graph $S(6)$ over a partition of $V$ into six classes, paged at Example 1.
• Example 2 (p. 324): $V$ is a set of $n$ points on the unit circle and the edges are the triples whose triangle contains the origin, with the tacit assumption that the origin lies in the convex hull of the points and on no line through two of them. The paper states that any four points of this 3-graph span zero or two edges.

Remark 1 (p. 327). The proof shows that in Example 2 the points can always be moved to the vertices of a regular $(2k+1)$-gon without changing the 3-graph; in particular, if every two vertices lie in a common edge, then $n$ is odd.

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 1 on p. 324, its proof in Section 3 (pp. 325--327), Remark 1 on p. 327. The edition is identified on the source card.

Read depth. Claims checked: the conventions, the statement, Example 2 and Remark 1 were read clause by clause on the page images. The proof was read but not checked step by step.

Proof pointer

Section 3 (pp. 325--327) splits on the vertex links $N(v)={xy:xyv\in\mathcal E}$. (a) If some link contains an odd cycle, a shortest one has length five (Proposition 1, p. 325), the vertex and that cycle span a copy of $S(6)$, and any further vertex extends it to a 3-graph of the form $H_S$; the general case follows by induction on $n$. (b) If every link is bipartite, the paper fixes a vertex $x$ and a bipartition $(A,B)$ of $N(x)$, orders each side by degree in $N(x)$, and proves Propositions 2--6 (pp. 326--327); after removing equivalent vertices (those with equal links, which by Proposition 6 are the pairs lying in no common edge), the degrees force $\lvert A\rvert=\lvert B\rvert=k$ and a regular $(2k+1)$-gon placement realizing Example 2.

Dependencies

Example 1 ($S(6)$ and $H_S$).

Bears on

• Problem 794: the problem page names Theorems 1--2 only as context, recording that they concern the stricter condition that every four vertices span exactly 0 or 2 edges; no other Erdős problem page in the corpus cites this paper. The theorem is the structural input to Theorem 2, whose bearing on that problem is recorded there.