OpenFrontiers
Join
← Back to the index
OP-9170OpenGeometry and topology

Erdős #757 · geometry / distances

Let A⊂RA\subset \mathbb{R} be a set of size nn such that every subset B⊆AB\subseteq A with ∣B∣=4\lvert B\rvert =4 has ∣B−B∣≥11\lvert B-B\rvert\geq 11. Find the best constant c>0c>0 such that AA must always contain a Sidon set of s…

Source reports OPENAs of 2026-09-08Source & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Let A⊂RA\subset \mathbb{R} be a set of size nn such that every subset B⊆AB\subseteq A with ∣B∣=4\lvert B\rvert =4 has ∣B−B∣≥11\lvert B-B\rvert\geq 11. Find the best constant c>0c>0 such that AA must always contain a Sidon set of size ≥cn\geq cn.

Lean 4

No Lean submitted

Formal statement

The formalization is available at the source below; its code is not attached here yet.

On-site verification needs an executable Lean target. None is attached yet.

A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.

Proofs & verification

No Lean has been submitted for this problem yet. A Lean file can be attached to a submission for review, though with no target here it cannot be machine-checked.

Share a formalization →Verification guide ↗

Background

For comparison, note that if BB were a Sidon set then ∣B−B∣=13\lvert B-B\rvert=13, so this condition is saying that at most one difference is 'missing' from B−BB-B. Equivalently, one can view AA as a set such that every four points determine at least five distinct distances, and ask for a subset with all distances distinct.

Without loss of generality, one can assume A⊂NA\subset \mathbb{N}.

Erdős and Sós proved that c≥1/2c\geq 1/2. Gyárfás and Lehel [GyLe95] proved

12+1141⋅76≤c≤35.\frac{1}{2}+\frac{1}{141\cdot 76}\leq c\leq\frac{3}{5}.

(The example proving the upper bound is the set of the first nn Fibonacci numbers.) Ma and Tang [MaTa26] have improved these bounds to

917≤c≤47.\frac{9}{17}\leq c\leq \frac{4}{7}.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #757 link
  2. [2]Upstream Lean 4 statement for Erdős #757 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 757 link

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.