Constructing integer sets in which a certain equation has no solution
Consider an equation
$$\tag{E}a_1x_1+\dots+a_kx_k=0.$$
Let $s=\sum_ia_i$ be the sum of its coefficients.

If $s=0$, the equation (E) has solutions in every nonempty set $S$, hence you are out of luck.

If $s\ne0$, let $p$ be the smallest prime not dividing $s$. Note that $p=O(\log |s|)$. Then (E) has no solution such that $x_i\equiv1\pmod p$ for all $i=1,\dots,k$, hence the set
$$S=\{1,p+1,\dots,(n-1)p+1\}$$
works, with maximum $(n-1)p+1$. For a fixed equation (E), this is $O(n)$, and the set can be computed as easily as it gets.