---

Statement

Theorem 7 (p. 304). $f_4^s(n) \le cn^{17/6}$.

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 7 on p. 304, proof pp. 304–305.

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

Proof pointer

Fix a non-degenerate triangle $ABC$ and form the 3-uniform hypergraph whose edges are the triples spanning a triangle similar to $ABC$. The paper argues that it contains no complete tripartite $K_3(2,3,3)$, since that would need three mutually orthogonal pieces (a line and two planes) fitting only in five or more dimensions, and cites the methods of Kővári–Sós–Turán and of Erdős (Israel J. Math. 2 (1964)) for the bound $cn^{3-1/(k\ell)}$ on the edges of a 3-graph with no $K_3(k,\ell,m)$, $c$ depending only on $k,\ell,m$ (p. 305). With $k=2$, $\ell=3$ this gives $cn^{17/6}$.

Bears on

No Erdős problem page of the corpus cites this result.