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Statement

Printed p. 62 = PDF p. 10, Section 5, read on the page image; unnumbered. After the quadratic order of $f^{(3)}(n;k,k-2)$ (Section 5 bound), the authors write: "We conjecture that $\lim_{n\to\infty} n^{-2}f^{(3)}(n;k,k-2)$ exists, but have succeeded [4] in proving this only for $k=4$." The conjecture asks only that the limit exist; it names no value. For $k=4$ the list on p. 58, from the authors' earlier paper [4], gives the value $\lim_{n\to\infty}n^{-2}f^{(3)}(n;4,2)=1/6$.

Source. W. G. Brown, P. Erdős and V. T. Sós, Some extremal problems on $r$-graphs, in New Directions in the Theory of Graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, 1971), Academic Press, New York (1973), 53--63, p. 62; the edition is identified in the source digest.

Proof pointer

A conjecture; the paper proves only the case $k=4$, in its reference [4].

Bears on

• Problem 1076: under the site's wording the problem asks whether this limit, for each $k\ge5$, equals $1/6$; the conjecture asks only that the limit exist, and the value is not part of it.
• Problem 1157: the conjecture concerns the problem's function in the case $r=3$, $k=s+2$, whose limit the problem page's quadratic-regime notes discuss.