formal-conjectures · erdos-problems · ams-5
Let $F(n,\alpha)$ denote the smallest $m$ such that there exists a $2$-colouring of the edges of
$K_n$ so that every $X\subseteq [n]$ with $\lvert X\rvert\geq m$ contains more than
$\alpha \binom{\lvert X\rvert}{2}$ many edges of each colour.
Prove that, for every $0\leq \alpha < 1/2$,
$$F(n,\alpha)\sim c_\alpha\log n$$
for some constant $c_\alpha$ depending only on $\alpha$.
This problem is #39 in Ramsey Theory in the graphs problem collection.
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/563
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/ErdosProblems/563.lean
Local target: Corpus.Erdos563.erdos_563
Source SHA-256: 602092c5f9f6ed1eb05420c0d23c91c3524b5c0218a1c346a402cf52b8bda3d2
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.
Let $F(n,\alpha)$ denote the smallest $m$ such that there exists a $2$-colouring of the edges of $K_n$ so that every $X\subseteq [n]$ with $\lvert X\rvert\geq m$ contains more than $\alpha \binom{\lvert X\rvert}{2}$ many edges of each colour.
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