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Statement

Theorem 4 (p. 299). $f_4^e(n) \le cn^{8/3}$.

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 4 on p. 299, proof pp. 299–300; the Remark after it on pp. 300–301.

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

Proof pointer

Fix a point $X_0$ and join $X_i$ to $X_j$ when $X_0X_iX_j$ is equilateral. The paper shows this graph contains no $K_{3,3}$ (pp. 299–300), so by the Kővári–Sós–Turán theorem it has fewer than $cn^{5/3}$ edges, and each point lies in at most $cn^{5/3}$ equilateral triangles.

The Remark (p. 300) states without full proof that the same argument, slightly elaborated, shows: for distinct points $X_1,\ldots,X_n$ in $E_4$ and an acute or obtuse triangle $XYZ$, no vertex lies in more than $cn^{5/3}$ triangles similar to $XYZ$. Its example (pp. 300–301), the origin with $n$ points on each of two circles of radius one about it in orthogonal coordinate planes, shows this fails for a right triangle: the origin lies in $n^2$ congruent isosceles right triangles.

Bears on

No Erdős problem page of the corpus cites this result.