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Statement

Theorem 10 (p. 306). $f_3^c(n) \le cn^{19/9}$.

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 10 on p. 306.

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

Proof pointer

Fix a non-degenerate triangle $ABC$. Erdős's bound $g_3(n)<c_2n^{5/3}$ on repeated distances in space leaves at most $cn^{5/3}$ pairs at distance $AB$; for each, the third vertices lie on a bounded number of circles, so there are $N\le cn^{5/3}$ circles. The counting of Theorem 6 then bounds the triangles by $2N+\{n(n-1)(n-2)\}^{1/3}N^{2/3}\le cn^{19/9}$.

Bears on

No Erdős problem page of the corpus cites this result.