  We consider Ramsey numbers $r(G,H)$ with tight lower bounds, namely,
\begin{align*} r(G,H) \geq (\chi(G)-1)(|H|-1)+1, \end{align*} where $\chi(G)$
denotes the chromatic number of $G$ and $|H|$ denotes the number of vertices in
$H$. We say $H$ is $G$-good if the equality holds.
  In this paper, we prove that the fan-graph $F_n=K_1 + n K_2$ is $K_p$-good if
$n\geq 27p^2$, improving previous tower-type lower bounds for $n$ due to Li and
Rousseau (1996). The join graph $G+H$ is defined 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. 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 $a_i \leq a_{i+1}$. 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 & \textrm{if $n|H|+a_2-1$ or $a_2-1$ is even,}\\ (p-1)(n
|H|+a_2-2)+1 & \textrm{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.
