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Statement

Theorem 8 (p. 305). $f_5^s(n) \le cn^{26/9}$.

In Section 3 (p. 301) the paper notes $f_5^e(n)\le f_5^s(n)\le cn^{26/9}$ for equilateral triangles in $E_5$; the second inequality is Theorem 8.

Notation. For distinct points $X_1,\ldots,X_n$ in $k$-dimensional Euclidean space $E_k$, the paper writes $f_k^i(n)$, $f_k^e(n)$, $f_k^c(n)$ and $f_k^s(n)$ for the largest possible number of isosceles triangles (congruent or not), of equilateral triangles, of pairwise congruent triangles and of pairwise similar triangles among them, the maximum taken over all choices of the $n$ points (pp. 291–292). Here $c$, $c_1$, $c_2$ are positive constants, not necessarily the same at each occurrence (p. 291).

Source. P. Erdős and G. B. Purdy, Some extremal problems in geometry, III, Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton, 1975), Congress. Numer. XIV, Utilitas Math., Winnipeg, 1975, pp. 291–308. The edition read is identified on the source card. Theorem 8 on p. 305.

Read depth. Claims checked: the statement was read clause by clause on the printed page.

Proof pointer

As for Theorem 7: the paper states that the 3-graph of triangles similar to a fixed triangle contains no $K_3(3,3,3)$, so it has fewer than $cn^{26/9}$ edges.

Bears on

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