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Erdős Problem #1087

OPENdatabase updated 2025-10-17

No proof or disproof is known, and it isn't known whether a finite computation could settle it.

What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).

A finite set of points has a repeated distance if two of its pairs of points are the same distance apart, that is, if it determines fewer distinct distances than it has pairs.

The number of degenerate four-element subsets of `P`, that is, those in which two of the six pairwise distances are equal.

\(f(n)\): the maximum, over all sets of \(n\) points in the plane, of the number of degenerate four-element subsets. The set of values is nonempty and bounded by \(\binom{n}{4}\), so `sSup` is a true maximum.

Let \(f(n)\) be minimal such that every set of \(n\) points in \(\mathbb{R}^2\) contains at most \(f(n)\) many sets of four points which are 'degenerate' in the sense that some pair are the same distance apart. Estimate \(f(n)\) - in particular, is it true that \(f(n)\leq n^{3+o(1)}\)? Here a set of four points is degenerate if two of its six pairwise distances are equal, and \(f(n)\leq n^{3+o(1)}\) is read as \(f(n) \ll n^{3+\varepsilon}\) for every \(\varepsilon > 0\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1087.
Prizenone
Topicsgeometry, distances
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1087

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