Erdős Problem 535

formal-conjectures · erdos-problems · ams-5 · ams-11

Let $r \geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\{1,\ldots,N\}$
such that no subset of size $r$ has the same pairwise greatest common divisor between all
elements. Erdős [Er64] proved that $f_3(N) > N^{c/\log\log N}$ for some constant $c > 0$, and
conjectured this should also be an upper bound; here we state the conjectural upper bound
for all $r \geq 3$.

See also [536].

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://www.erdosproblems.com/535
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/ErdosProblems/535.lean
Local target: Corpus.Erdos535.erdos_535
Source SHA-256: 353e59057cf7a50fabeece2aec2bed077f05a7075084d0ac839cf215e29579c9
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record