This question is motivated by Erdős conjecture on arithmetic progressions. It is a weaker conjecture than Erdős' one.
Erdős conjecture on arithmetic progressions states, "If $A$ is a large set, then $A$ contains (nontrivial) arithmetic progressions of every length."
The following conjecture states that, "If $A$ is a large set, then $A$ contains (nontrivial) polynomial progressions of integer coefficients of every length."
Conjecture: If $A\subset \mathbb{N}$ is a large set in the sense that
$$ \sum_{n\in A} \frac{1}{n} = \infty,$$
then for every $k\in\mathbb{N},$ there exists a polynomial of integer coefficients $(c_i)_{i=1}^{n}\ $ with $\ n\geq 1\ $ and $\ c_n\geq 1$,
$$ f_k(x) = c_0 + c_1 x + c_2 x^2 + \ldots + c_n x^n, $$
such that
$$ f_k(i) \in A \quad \forall\ i\in \lbrace{1,\ldots, k\rbrace}. $$
Am I missing something and somehow this is relatively easily proved using elementary number theory, or would you expect this to be almost as difficult as Erdős conjecture itself?
In fact this can be done rather trivially for any infinite set $A$ of integers.
Given $k$, let $$L (x) = \prod_{j=2}^k \frac{x - j}{1-j}$$ a polynomial of degree $k-1$ interpolating $(1,1)$ and $(j,0)$ for $j = 2 \ldots k$. Its coefficients are integers divided by $(k-1)!$. Let $y_1$ and $y_2$ be two members of $A$ that are congruent mod $(k-1)!$. Then the polynomial
$$f_k(x) = y_1 + (y_2 - y_1) L(x)$$
has degree $k-1$, integer coefficients $f_k(1) = y_2$ and $f_k(j) = y_1$ for $j = 2 \ldots k$. If the leading coefficient is negative, interchange $y_1$ and $y_2$ to make it positive.
It's also easy to modify this to get a solution where the values $f_k(j)$ for $j = 1, \ldots, k$ are distinct.
[EDIT] Consider the polyomials $$L_i(x) = \prod_{j \in \{1,\ldots,k\} \backslash \{i\}} \frac{x-j}{i-j}$$ This is the unique polynomial of degree $k-1$ with value $1$ at $i$ and $0$ at the other integers from $1$ to $k$. Note $\sum_{i=1}^k L_i(x) = 1$. Then $$f(x) = \sum_{i=1}^k y_i L_i(x)$$ has $f(i) = y_i$ for $i = 1, \ldots, k$. If all the $y_i$ are congruent modulo the lcm of the denominators, this polynomial has integer coefficients.