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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 $k$-connectedmonochromatic subgraph of order at least $n-f(G,k,m)$ in any rainbow $G$-freecoloring of $K_n$ using all the $m$ colors. In this paper, we prove that theset $\mathcal{A}$ consists of precisely $P_6$, $P_3\cup P_4$, $K_2\cup P_5$,$K_2\cup 2P_3$, $2K_2\cup K_3$, $2K_2\cup P^{+}_4$, $3K_2\cup K_{1,3}$ andtheir 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 $c=c(H)$ isa 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 $H$ oforder at least 3 such that there is a $k$-connected monochromatic subgraph oforder 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 $t$. Weprove that the set $\mathcal{B}$ consists of precisely $2P_3$, $2K_2\cupK_{1,3}$ and their subgraphs of order at least 3.  Finally, we consider the large $k$-connected multicolored subgraph instead ofmonochromatic subgraph. We show that for $1\leq k \leq 3$ and $n$ sufficientlylarge, every Gallai-3-coloring of $K_n$ contains a $k$-connected subgraph oforder at least $n-\left\lfloor\frac{k-1}{2}\right\rfloor$ using at most twocolors. We also show that the above statement is false for $k=4t$, where $t$ isan positive integer.

Forbidden rainbow subgraphs that force large monochromatic or multicolored k-connected subgraphs