Problem 812

Erdős Problem #812: declared status 'open'. Formalized: yes. 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$$R(n+1)-R(n) \gg n^2?$$ Current best: Burr, Erd\H{o}s, 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$$R(n+2)-R(n) \gg n^{2-o(1)}.$$ References [BEFS89] Burr, S. Prize: no. OEIS: A059442. Tags: graph theory, ramsey theory.