Growth rate of diagonal Ramsey numbers

Open

erdos-ramsey-growth

Statement

For the diagonal Ramsey number

R(k)

, determine whether

\lim_k R(k)^{1/k}

exists and its value. It is known that

\sqrt{2} \leq \liminf \leq \limsup \leq 4

Current frontier

erdosproblems.com/77, OPEN. Lower bound

\sim (k/(e\sqrt2))\,2^{k/2}

(Erdős 1947); Campos–Griffiths–Morris–Sahasrabudhe (2023) gave the first exponential upper-bound gain (

\varepsilon \approx 2^{-7}

), since lowered to

R(k) \leq 3.7992^k

by Gupta–Ndiaye–Norin–Wei (2024).

When this counts as solved

VALUE-DETERMINATION. PROOF_COMPLETE for the limit and its value with matching bounds. BREAKTHROUGH for raising the lower-bound base above

\sqrt 2

, lowering the upper base below the current best

3.7992

, or proving the limit exists.

Classification

value-determination