Let $\mathcal I$ be the set of increasing injections from $\mathbb N$ to $\mathbb N$, and $\alpha$ a well ordered uncontable set that ordinal type is regular. Suppose that $f : \beta \mapsto f_{\beta}$ is a fonction from $\alpha$ to $\mathcal I$ such that for any $(i,j)\in \alpha\times \alpha$, such that $i<j$, there exists $k\in \mathbb N$ such that for all $n>k$,
$$f_i(n)<f_j(n)\tag{1}$$
Now suppose that there exists $g\in \mathcal I$ such that $f_i(n)<g(n)$ for all $i\in \alpha$ and $n\in \mathbb N$.
Question.
Does there exists an infinite $A\subset \mathbb N$ and $\gamma\subset \alpha$ such that $|\gamma|=|\alpha|$ and such that for any $c\in \gamma$, $|A\setminus f_c(\mathbb N)|<\infty$.
(I'm not assuming $CH$)
Note about the title : The condition $(1)$ gives a quasi order on $\mathcal I$, according to whom I use the words "decreasing" and "low-bounded" in the title.