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Statement

Theorem 6 (p. 302). $f_3^s(n) \le cn^{7/3}$.

In Section 3 (p. 299) the paper notes $f_3^e(n)\le f_3^s(n)\le cn^{7/3}$ for equilateral triangles in space; the second inequality is Theorem 6.

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 6 on p. 302, proof pp. 302–303.

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

Proof pointer

Fix a non-degenerate triangle $ABC$. For each pair $X_i, X_j$ the third vertices of triangles similar to $ABC$ lie on a bounded number of circles, so there are $N\le cn^2$ circles. Three points lie on at most one circle, so $\sum\binom{v_i}{3}\le\binom n3$ for the numbers $v_i$ of points on the circles, and convexity bounds $\tfrac13\sum v_i$ by $\tfrac23N+\tfrac13\{n(n-1)(n-2)\}^{1/3}N^{2/3}\le cn^{7/3}$ (p. 303).

Bears on

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