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Erdős problem 120 Let A ⊆ R A \subseteq \mathbb{R} A ⊆ R be an infinite set. Must there be a set E ⊆ R E \subseteq \mathbb{R} E ⊆ R
of positive measure which does not contain any set of the shape a ∗ A + b a * A + b a ∗ A + b
for some
a , b ∈ R a,b \in \mathbb{R} a , b ∈ R and
?
$4,011 bounty · 1 piece from 1 person· last one 15 days ago
week of 1 Oct 2025 · nothing published week of 8 Oct 2025 · nothing published week of 15 Oct 2025 · nothing published week of 22 Oct 2025 · nothing published week of 29 Oct 2025 · nothing published week of 5 Nov 2025 · nothing published
Solve it A complete proof or refutation in Lean. The first one the validator accepts takes the whole bounty.
Add a piece A lemma, a definition, a special case. There is no race: when someone finishes the problem, the pool splits between everyone whose work led there.
Command line, then a pull request on GitHub.
← Erdős problem 108 Erdős problem 124 - ne zero → What has been built 1 piece · 15 lemmas
8 Sept 2026
Disproof target
import FormalConjectures.ErdosProblems.«120»
import TaskSupport
namespace Bounty
theorem target : ¬ (fcTypeOfName% "Erdos120.erdos_120") := by
sorry
end Bounty
Open this target’s Challenge.lean Original conjecture (Lean type)
True ↔ ∀ (A : Set ℝ), A.Infinite → Erdos120.Erdos120For AOriginal conjecture source: FormalConjectures/ErdosProblems/120.lean
Source type SHA-256 sha256:3f7260d6fda3c348905c579e440487208297a8a0db2d9f4ab48aaf196499bf2c
Task id fc-6a786f99-erdos120-erdos-120-00a48cdedb-counterexample-v1
Task commitment sha256:bcadb3c5c04cd82927002baa7a223f42b0d373ddc82ce5f335fcee5f4e18790e week of 12 Nov 2025 · nothing published
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1 Oct 2025 a year of work on this problem 23 Sept 2026
erdosproblems.com/120
Something wrong with this formalization?
A statement that does not faithfully capture the original conjecture is the one real risk here, so we would rather hear about it early - before someone spends weeks on it.
Our channel is in the Bittensor Discord server. Join the server first, then open the channel to send your report.
Reward
$4,011
for closing this problem
Whoever closes it takes the bounty. Published pieces that helped get there share a separate contribution pool, paid out once the problem is closed.
Set by bounty policy linear-age-v3-locked: the amount is worked out from how long the problem has stood open, so it moves as the pool and the pool's age profile move.
Prove it or break it
The same problem accepts both. You can submit a proof of the statement or a counterexample that refutes it, and you choose which when you submit.
Prove the statement no attempts
Find a counterexample no attempts
Erdős problem 120 · Conjectures.io