Fan-complete Ramsey numbers

For graphs $G$ and $H$, we consider Ramsey numbers $r(G,H)$ with tight lower bounds, namely, $r(G,H)\ge (\chi (G)-1)(\mathrm{\mid }H\mathrm{\mid }-1)+1,$ where $\chi (G)$ denotes the chromatic number of $G$ and $\mathrm{\mid }H\mathrm{\mid }$ denotes the number of vertices in $H$. We say $H$ is $G$-good if the equality holds. Let $G+H$ be the join graph obtained from graphs $G$ and $H$ by adding all edges between the disjoint vertex sets of $G$ and $H$. Let $nH$ denote the union graph of $n$ disjoint copies of $H$. We show that $K_1+nH$ is $K_p$-good if $n$ is sufficiently large. In particular, the fan-graph $F_n=K_1+n{K}_2$ is $K_p$-good if $n\ge 27p^2$, improving previous tower-type lower bounds for $n$ due to Li and Rousseau (1996). Moreover, we give a stronger lower bound inequality for Ramsey number $r(G,K_1+F)$ for the case of $G=K_p(a_1,a_2,\dots ,a_p)$, the complete $p$-partite graph with $a_1=1$ and $a_i\le a_{i+1}$. In particular, using a stability-supersaturation lemma by Fox, He and Wigderson (2021), we show that for any fixed graph $H$, \begin{align*} r(G,K_1+nH) = \left{ \begin{array}{ll} (p-1)(n |H|+a_2-1)+1 & $\text{if }n\mathrm{\mid }H\mathrm{\mid }+a_2-1\text{ is even or }{a}_2-1\text{ is even,}$\ (p-1)(n |H|+a_2-2)+1 & $\text{otherwise,}$ \end{array} \right. \end{align*} where $G=K_p(1,a_2,\dots ,a_p)$ with $a_i$’s satisfying some mild conditions and $n$ is sufficiently large. The special case of $H=K_1$ gives an answer to Burr’s question (1981) about the discrepancy of $r(G,K_{1,n})$ from $G$-goodness for sufficiently large $n$. All bounds of $n$ we obtain are not of tower-types.