Problem Statement
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\[R(n+1)-R(n) \gg n^2?\]
Categories:
Graph Theory Ramsey Theory
Progress
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\[R(n+2)-R(n) \gg n^{2-o(1)}.\]Source: erdosproblems.com/812 | Last verified: January 16, 2026