Erdős Problems

SOLVED

Prove that \[R(C_k,K_n)=(k-1)(n-1)+1\] for $k\geq n\geq 3$ (except when $n=k=3$).

Asked by Erdős, Faudree, Rousseau, and Schelp, who also ask for the smallest value of $k$ such that this identity holds (for fixed $n$). They also ask, for fixed $n$, what is the minimum value of $R(C_k,K_n)$?

This identity was proved for $k>n^2-2$ by Bondy and Erdős [BoEr73]. Nikiforov [Ni05] extended this to $k\geq 4n+2$.

Keevash, Long, and Skokan [KLS21] have proved this identity when $k\geq C\frac{\log n}{\log\log n}$ for some constant $C$, thus establishing the conjecture for sufficiently large $n$.