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Statement

As printed on p. 78 (PDF p. 4 of the typescript scan, page image): "Let $G$ be the skeleton of a cube. Simonovits and I proved [9]

$$f(n;G)<cn^{8/5} \tag{7}$$

We could not decide whether (7) is best possible."

The skeleton of a cube is the graph $Q_3$ of its vertices and edges; $f(n;G)$ is the smallest number of edges forcing $G$, so (7) is $\mathrm{ex}(n;Q_3)<cn^{8/5}$. The paper's [9] is the 1970 Balatonfüred paper of Erdős and Simonovits, whose display (5) proves the bound (equation_5). The same page (display (6) and the sentences before it) records the disproved conjecture that the exponent of every bipartite graph has the form $1+1/k$ or $2-1/k$ and the surviving conjecture that $\lim f(n;G)/n^\alpha=c(G)$ exists for some $\alpha\in(1,2)$, "Probably the $\alpha$ in (6) is always rational"; these belong to Problem 713.

Source. P. Erdős, Extremal problems on graphs and hypergraphs, Hypergraph Seminar, Lecture Notes in Math. 411 (1974), 75--84; printed p. 78 = PDF p. 4 of the ten-page typescript scan (printed p. $n$ = PDF p. $n-74$), read on the rendered page image. The artifact is identified in the source digest.

Read depth. Claims checked: the display and its two sentences were read clause by clause on the page image. The paper gives no proof.

Proof pointer

None in the source; see the 1970 paper's display (5).

Dependencies

None stated.

Bears on

• Problem 576: the site's [Er74c, p. 78] source; the upper bound and the question "whether (7) is best possible", the form in which Erdős asked the cube problem in 1974.