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Source. Section 4, p. 251, of Paul Erdős and George Purdy, Some extremal problems in geometry, J. Combinatorial Theory 10 (1971), no. 3, 246--252, DOI 10.1016/0097-3165(71)90028-8, as identified on the source card. The statement is unnumbered; this page takes its name from the page.

Statement

Question (p. 251). Given $n$ points in the plane, how many four-point subsets can have six mutual distances that are not all different?

The paper then states, without proof (p. 251), three things about the maximum number of such quadruples:

• that it is not difficult to show that $n$ points can be placed so that there are $cn^3\log n$ such quadruples;
• that one "cannot have $cn^{7/2}$ such quadruplets", with $c$ not further specified (p. 251);
• that "It seems that the maximum is less than $n^{3+\epsilon}$ but we could not prove this" (p. 251).

No construction or proof of the first two is printed in the paper.

Proof pointer

None in the paper: the section introduces "related combinatorial problems" (p. 251) and gives neither the construction nor the upper-bound argument.

Dependencies

None. Read depth: claims checked; the paragraph was read clause by clause on p. 251.

Bears on

• Problem 1087: this paragraph is the question that problem states, a four-point set being degenerate when some two of its six distances are equal. The paper asserts, without printing a proof, a construction with $cn^3\log n$ such sets and an upper bound below $cn^{7/2}$, and states as its belief, not proved, the bound $n^{3+\epsilon}$ that the problem asks about.