All Random Solved Random Open
OPEN - $250
Find the value of $\lim_{k\to \infty}R(k)^{1/k}$.
Erdős offered \$100 for just a proof of the existence of this constant, without determining its value. He also offered \$1000 for a proof that the limit does not exist, but says 'this is really a joke as [it] certainly exists'. Erdős 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].

See also the entry in the graphs problem collection.