Star-critical Ramsey numbers involving large generalized books

Abstract

For graphs $F$, $G$, and $H$, we write $F\to (G,H)$ if every red/blue edge-coloring of $F$ contains either a red copy of $G$ or a blue copy of $H$. The Ramsey number $R(G,H)$ is the smallest integer $N$ such that $K_N\to (G,H)$. Let $H:=K_k+n{K}_h$ be the generalized book graph, and let $G:=K_{p+1}(a_1,a_2,\dots ,a_{p+1})$ be a complete $(p+1)$-partite graph satisfying $a_1=1$, $a_2\mid (nh-1)$, and $a_i\le a_{i+1}$. In this paper, avoiding the use of Szemerédi's regularity lemma, we prove that for any fixed $h,p\ge 1$, $k\ge 2$, and sufficiently large $n$, $K_{p(nh+a_{2}k-1)+1}\setminus K_{1,nh-a_2(h-2)-1}\to (G,H).$ This result yields the star-critical Ramsey number $r_{\ast }(G,H)=(p-1)(nh+a_{2}k-1)+a_2(k+h-2)+1.$