  Let $n, k, m$ be positive integers with $n\gg m\gg k$, and let $\mathcal{A}$
be the set of graphs $G$ of order at least 3 such that there is a $k$-connected
monochromatic subgraph of order at least $n-f(G,k,m)$ in any rainbow $G$-free
coloring of $K_n$ using all the $m$ colors. In this paper, we prove that the
set $\mathcal{A}$ consists of precisely $P_6$, $P_3\cup P_4$, $K_2\cup P_5$,
$K_2\cup 2P_3$, $2K_2\cup K_3$, $2K_2\cup P^{+}_4$, $3K_2\cup K_{1,3}$ and
their subgraphs of order at least 3. Moreover, we show that for any graph $H\in
\mathcal{A}$, if $n$ sufficiently larger than $m$ and $k$, then any rainbow
$(P_3\cup H)$-free coloring of $K_n$ using all the $m$ colors contains a
$k$-connected monochromatic subgraph of order at least $cn$, where $c=c(H)$ is
a constant, not depending on $n$, $m$ or $k$.
  Furthermore, we consider a parallel problem in complete bipartite graphs. Let
$s, t, k, m$ be positive integers with ${\rm min}\left\{s, t\right\}\gg m\gg k$
and $m\geq |E(H)|$, and let $\mathcal{B}$ be the set of bipartite graphs $H$ of
order at least 3 such that there is a $k$-connected monochromatic subgraph of
order at least $s+t-f(H,k,m)$ in any rainbow $H$-free coloring of $K_{s,t}$
using all the $m$ colors, where $f(H,k,m)$ is not depending on $s$ or $t$. We
prove that the set $\mathcal{B}$ consists of precisely $2P_3$, $2K_2\cup
K_{1,3}$ and their subgraphs of order at least 3.
  Finally, we consider the large $k$-connected multicolored subgraph instead of
monochromatic subgraph. We show that for $1\leq k \leq 3$ and $n$ sufficiently
large, every Gallai-3-coloring of $K_n$ contains a $k$-connected subgraph of
order at least $n-\left\lfloor\frac{k-1}{2}\right\rfloor$ using at most two
colors. We also show that the above statement is false for $k=4t$, where $t$ is
an positive integer.
