Problem 77

Erdős Problem #77: declared status 'open'. Formalized: no. If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, then find the value of$$\lim_{k\to \infty}R(k)^{1/k}.$$ Current best: Erd\H{o}s proved$$\sqrt{2}\leq \liminf_{k\to \infty}R(k)^{1/k}\leq \limsup_{k\to \infty}R(k)^{1/k}\leq 4.$$The upper bound has been improved to $4-\tfrac{1}{128}$ by Campos, Griffiths, Morris, and Sahasrabudhe [CGMS23]. A shorter and simpler proof of an upper bound of the strength $4-c$ for some constant $c>0$ (and a generalisation to the case of more than two colours) was given by Balister, Bollob\'{a}s, Campos, Griffiths, Hurley, Morris, Sahasrabudhe, and Tiba [BBCGHMST24]. See also [1029] for a problem concerning a lower bound for $R(k)$ and discussion of lower bounds in general. and Wei, L., Optimizing the CGMS upper bound on Ramsey numbers. Prize: $250. OEIS: A059442. Tags: graph theory, ramsey theory.