Star-critical Ramsey numbers involving large books

Abstract

For graphs $F,G$ and H, let $F\to (G,H)$ signify that any red/blue edge coloring of F contains either a red G or a blue H. The Ramsey number $r(G,H)$ is defined to be the smallest integer r such that $K_r\to (G,H)$. Let $B_n^{\left(k\right)}$ be the book graph which consists of n copies of $K_{k+1}$ all sharing a common $K_k$, and let $G:=K_{p+1}(a_1,a_2,...,a_{p+1})$ be the complete $(p+1)$-partite graph with $a_1=1$, $a_2\left|\right(n-1)$ and $a_i\le a_{i+1}$.

In this paper, avoiding the use of Szemerédi's regularity lemma, we show that for any fixed $p\ge 1$, $k\ge 2$ and sufficiently large n, $K_{p(n+a_{2}k-1)+1}\setminus K_{1,n+a_2-2}\to (G,B_n^{\left(k\right)})$. This implies that the star-critical Ramsey number $r_{⁎}(G,B_n^{\left(k\right)})=(p-1)(n+a_{2}k-1)+a_2(k-1)+1$. As a corollary, $r_{⁎}(G,B_n^{\left(k\right)})=(p-1)(n+k-1)+k$ for $a_1=a_2=1$ and $a_i\le a_{i+1}$. This solves a problem proposed by Hao and Lin (2018) [11] in a stronger form.