OPEN
Is it true that
\[\frac{R(n+1)}{R(n)}\geq 1+c\]
for some constant $c>0$, for all large $n$? Is it true that
\[R(n+1)-R(n) \gg n^2?\]
Burr, Erdős, Faudree, and Schelp
[BEFS89] proved that
\[R(n+1)-R(n) \geq 4n-8\]
for all $n\geq 2$. The lower bound of
[165] implies that
\[R(n+2)-R(n) \gg n^{2-o(1)}.\]
