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Statement

For integers $n>k>r$ and $0<s\le\binom kr$, $m(n,r,k,s)$ is "the maximum number of edges in an $r$-graph on $n$ vertices in which any $k$ vertices span less than $s$ edges" (p. 323); so $m(n,3,4,3)$ is the largest number of triples on $n$ points with no four points spanning three triples, the Turán number of the 3-graph on four vertices with three edges. Theorem 3 (p. 325):

$$\frac{2+o(1)}{7}\binom n3\le m(n,3,4,3)\le\frac13\binom n3\frac{n}{n-2}.$$

"The upper bound was proved by Caen [2]. Let us mention that the lower bound of the theorem was proved independently by Giraud [5] also" (p. 325; [2] is D. de Caen, Ars Combinatoria 16 (1983) 5--10, and [5] "G. Giraud, Private communication by Paul Erdös", p. 328). The lower bound comes from Section 2, "A remark on $m(n,3,4,3)$" (pp. 324-325): once four points may span up to two edges rather than exactly zero or two, edges can be added to the six-class blow-up $H_S$ of Example 1 iteratively, by splitting each class into six sets, adding the triples that follow the pattern $S(6)$ across those sets, and repeating inside each set; "Making the partitions always as equal as possible, finally one obtains $n^3(1+o(1))/21$ edges." Since $\binom n3=(1+o(1))n^3/6$, that is $(2+o(1))\binom n3/7$. The introduction (pp. 323-324) records the target: "Turàn (cf. [3, 7]) conjectured that $m(n,3,4,3)$ is asymptotic to $n^3/24$", and already $H_S$ with $|V_i|\ge\lfloor n/6\rfloor$ "has more than $10\lfloor n/6\rfloor^3$ edges which is more than $n^3/24$, disproving Turàn's conjecture".

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 (the Crossref record); Theorem 3 on printed p. 325 = PDF p. 3 of the six-page publisher scan (printed p. $n$ = PDF p. $n-322$), read on the page image, with pp. 323-324 and 328 read for the definition, the conjecture and the references. The artifact is identified in the source digest.

Read depth. Claims checked: the statement, the definition of $m(n,r,k,s)$, the Section 2 construction paragraph and the attribution sentences were read clause by clause on the page images. The count $n^3(1+o(1))/21$ was not replayed, and de Caen's proof of the upper bound is not in the paper.

Proof pointer

Lower bound: the iterated construction of Section 2 (pp. 324-325), starting from Example 1's $H_S$ (the blow-up of the ten-triple 3-graph $S(6)$ on six points, p. 323). Upper bound: de Caen (reference [2]), not held.

Dependencies

De Caen's upper bound (external, not held); the six-point 3-graph $S(6)$ of p. 323 and Example 1.

Bears on

• Problem 794: the site's commentary reads the problem as asking for the density of $m(n,3,4,3)$, and the problem page records that reading as a variant with its own answer, not as a corrected statement. For that variant the lower bound gives the site's $2/7$, which the site calls the conjectured truth, the p. 324 disproof sentence refutes Turán's $n^3/24$ (density $1/4$), and the upper bound stated here is de Caen's $1/3$. The iterated construction also has more than $n^3+1$ triples on $3n$ points for large $n$, which the problem's claim page for this paper uses against the statement as printed.