Problem 135

Let $A\subset \mathbb{R}^2$ be a set of $n$ points such that any subset of size $4$ determines at least $5$ distinct distances. Must $A$ determine $\gg n^2$ many distances?

Source: erdosproblems.com/135

Distances
Geometry

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SolvedDisproved

An accepted solution exists. The statement is false.

Disproved. The site's export of 2026-09-04 records the label "DISPROVED (LEAN)". The disproof is Tao's construction, recorded on its claim page (Tao, 2024); the Lean qualification refers to a development in Boris Alexeev's repository that was not built here (see Formalization).

Er97b: Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. 165/166 (1997), 227--231, DOI 10.1016/S0012-365X(96)00173-2; item 12, printed p. 231 (PDF p. 5 of the publisher's open-archive file at that DOI): the conjecture of $c_1m^2$ distinct distances with an offer, and the stronger conjecture of $c_2m$ points with all distances distinct. Library home: Erdős 1997.
Ta24c: T. Tao, Planar point sets with forbidden 4-point patterns and few distinct distances. arXiv:2409.01343 (2024).

Problem 135

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