Diagonal Ramsey numbers

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $k\geq2$, $R(k,k)$ is the least integer $N$ such that every red--blue colouring of the edges of the complete graph $K_N$ contains a monochromatic copy of $K_k$.

Statement recorded in the supplied source.

Does the real limit \[L=\lim_{k\to\infty}R(k,k)^{1/k}\] exist? If it exists, determine its value. [S2]

Short English statement

Do the diagonal Ramsey numbers have a well-defined exponential growth rate, and what is that rate?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3069 (OPG-351). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Erdos, Paul, Diagonal Ramsey numbers, Open Problem Garden, node 351. Complete problem, discussion and bibliography (including mathematical image alt text). https://www.openproblemgarden.org/op/diagonal_ramsey_numbers.