Rainbow Turán problems for a matching and any other graph

For a family of graphs $\text{cF}$, a graph is called $\text{cF}$-free if it does not contain any member of $\text{cF}$ as a subgraph. Given a collection of graphs $(G_1,\dots ,G_t)$ on the same vertex set $V$ of size $n$, a rainbow graph on $V$ is obtained by taking at most one edge from each $G_i$. We say that a collection is rainbow $\text{cF}$-free if it contains no rainbow copy of any member of $\text{cF}$. In this paper, we study the maximum values of $mi{n}_{i\in [t]}\mathrm{\mid }E(G_i)\mathrm{\mid }$, ${\sum }_{i=1}^t\mathrm{\mid }E(G_i)\mathrm{\mid }$ and ${\prod }_{i=1}^t\mathrm{\mid }E(G_i)\mathrm{\mid }$ among rainbow $\{F,M_{s+1}\}$-free collections $(G_1,\dots ,G_t)$ on $n$ vertices.