Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 135

claims/: The 1 claim page of Problem 135, one per claimant's result; the problem's standing derives from them.

Statement. 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?

Status. 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; the Lean qualification refers to a development in Boris Alexeev's repository that was not built here (see Formalization).

Source. erdosproblems.com/135, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #135, https://www.erdosproblems.com/135.

References.

[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: erdos_1997_some_old_new_problems_various_branches_combinatorics.
[Ta24c] T. Tao, Planar point sets with forbidden 4-point patterns and few distinct distances. arXiv:2409.01343 (2024).

Formalization. No formal-conjectures statement is recorded; the site's page, lists none. A Lean proof of the disproof in Boris Alexeev's repository is linked at a pinned commit on the Tao claim page; this corpus has not built or audited it.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (11)

Tao's few-distance sets with five distances in every four points problems/distance_problems/E0135/claims Distance Problems Distance Problems distance_problems/fox_2016_more_distinct_distances_local_conditions distance_problems/sheffer_2014_distinct_distances_open_problems_current_bounds Problem 31 (p. 14): the asymptotic value of phi(n,4,5)distance_problems/tao_2024_planar_point_sets_forbidden_4_point Theorem 1.2 (Main theorem): grid subsets of size order n avoiding all eight four-point patterns Theorem 1.3 (Main theorem with survivors): grid subsets of size order n with O(n) copies of each pattern integer_sequences/erdos_1997_some_old_new_problems_various_branches_combinatorics

Graph