Problem Statement
If $R(k,l)$ is the Ramsey number then prove the existence of some $c>0$ such that\[\lim_k \frac{R(k+1,k)}{R(k,k)}> 1+c.\]
Categories:
Graph Theory Ramsey Theory
Progress
A problem of Erdős and Sós, who could not even prove whether $R(k+1,k)-R(k,k)>k^c$ for any $c>1$.It is trivial that $R(k+1,k)-R(k,k)\geq k-2$. Burr, Erdős, Faudree, and Schelp [BEFS89] proved\[R(k+1,k)-R(k,k)\geq 2k-5.\]See also [544] for a similar question concerning $R(3,k)$, and [1014] for the general off-diagonal case.
Source: erdosproblems.com/1030 | Last verified: January 19, 2026