  Given two graphs $G_1$ and $G_2$, the Ramsey number $r(G_1,G_2)$ refers to
the smallest positive integer $N$ such that any graph $G$ with $N$ vertices
contains $G_1$ as a subgraph, or the complement of $G$ contains $G_2$ as a
subgraph. A connected graph $H$ is said to be $p$-good if
$r(K_p,H)=(p-1)(|H|-1)+1$. A generalized fan, denoted as $K_1+nH$, is formed by
the disjoint union of $n$ copies of $H$ along with an additional vertex that is
connected to each vertex of $nH$. Recently Chung and Lin proved that $K_1+nH$
is $p$-good for $n\ge cp\ell/|H|$, where $c\approx 52.456$ and
$\ell=r(K_{p},H)$. They also posed the question of improving the lower bound of
$n$ further so that $K_1+nH$ remains $p$-good.
  In this paper, we present three different methods to improve the range of
$n$. First, we apply the Andr\'asfai-Erd\H{o}s-S\'os theorem to reduce $c$ from
$52.456$ to $3$. Second, we utilize the approach established by Chen and Zhang
to achieve a further reduction of $c$ to $2$. Lastly, we employ a new method to
bring $c$ down to $1$. In addition, when $K_1+nH$ forms a fan graph $F_n$, we
can further obtain a slightly more refined bound of $n$.
