Ramsey number of a path and $C_n^k$

Abstract:

Define the $k$-th power $F_k$ of a graph $F$ as a graph on $V (F)$, in which two vertices are adjacent if their distance in $F$ is at most $k$. Given graphs $G$ and $H$, Ramsey number $R(G,H)$ is the smallest integer $N$ such that any red-blue edge-coloring of $K_N$ contains a red copy of $G$ or a blue copy of $H$. Recently, Pokrovskiy proved that $R(P_n,P_n^k)=(n-1)k \lfloor \frac{n}{k 1}\rfloor$, which solves a conjecture of Allen, Brightwell and Skokan. In this paper, we show that $R(P_n,C_n^k)=(n-1)(\chi(C_n^k)-1) \sigma(C_n^k) o(n)$ holds for fixed $k$ and $n\to \infty$.