$P_n$和$C_n^k$的Ramsey数

Ramsey number of a path and $C_n^k$

摘要:

称 ~$F^k$~ 为图~$F$ ~的~$k$-幂次图，如果~$V(F^k)=V(F)$~, 且~$F^k$~中的任意两个顶点相邻当且仅当他们在~$F$~中的距离至多为~$k$。 给定图~$G$~和~$H$，~Ramsey~数~$R(G,H)$~为最小的正整数~$N$，使得完全图~$K_N$~的任意红蓝-边着色都会含有一个红色的子图 ~$G$~或者蓝色的子图~$H$。Pokrovskiy~证明了~$R(P_n,P_n^k)=(n-1)k \lfloor \frac{n}{k 1}\rfloor$，解决了~Allen~等人在~2010~年提出的一个猜想。本文证明渐近阶~$R(P_n,C_n^k)=(n-1)(\chi(C_n^k)-1) \sigma(C_n^k) o(n)$, 其中~$k$~是常数。

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 $n\to \infty$.