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Statement

Theorem 3 (p. 296). $\tfrac16 n^2 - cn^{3/2} \le f_2^e(n) \le n^2/3$.

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 3 on p. 296, proof pp. 296–299.

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

Proof pointer

Upper bound (p. 296): two points are vertices of at most two equilateral triangles, so $f_2^e(n)\le\tfrac23\binom n2$. Lower bound (pp. 296–299): the points of a scaled triangular lattice in the unit disc, numbering $n+O(\sqrt n)$; the lattice is closed under completing equilateral triangles, and integrating the area of overlap of two unit discs over the disc (the integral is $\pi^2/2$) counts $n^2/6-cn^{3/2}$ triangles. The Conclusion (p. 307) asks for $\lim_{n\to\infty} f_2^e(n)/n^2$, whether it exists, and whether $f_2^e(n)\le(\tfrac13-\epsilon)n^2$.

Bears on

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