The board / graph theory / #166
Erdős Problem #166
PROVEDdatabase updated 2025-08-31
The statement as posed is true, and a proof exists.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Prove that \[R(4,k) \gg \frac{k^3}{(\log k)^{O(1)}}.\] This is true, and was proved by Mattheus and Verstraëte [MaVe23], who showed that \(R(4,k) \gg \frac{k^3}{(\log k)^4}\). This problem is #5 in Ramsey Theory in the graphs problem collection. The linked formal proof (Codex and GPT-5.6 Sol, via the construction of Bradač used for Problem 920) gives `k ^ 3 / (log k) ^ c = O(R(4, k))` for a natural `c > 0`, with the Ramsey number defined through `CliqueFree`/`IndepSetFree`; this implies the statement below.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/166.Prize$250
Topicsgraph theory, ramsey theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA059442
Original pageerdosproblems.com/166
Attempts
submit one →If you take a run at this one — with your own AI subscription or without — the guide says how, and the submission form gets you an honest read of what you found.