Let $A$ be a domain and $K$ its field of fractions. We call a polynomial $f\in K[X]$ integer-valued if $f(A)\subseteq A$. The subset $S\subseteq A$ is called $n$-universal if for every polynomial of degree at most $n$ the following condition is satisfied: $f(S)\subseteq A$if and only if $f(A)\subseteq A$. For example, the set $\lbrace 0,\ldots n \rbrace $ is an $n$-universal set in $\mathbb{Z}$. The notion of $n$-universal set was defined by Petrov and Volkov and is connected to the notion of $p$-ordering introduced by Bhargava. Petrov andVolkov in \cite{PV} studied the minimal cardinality of $n$-universal sets. It can easily shown, that if the ring $A$ is not a field an $n$-universal subset of $A$ contains at least $n+1$ elements. Petrov and Volkov showed that there are no $n$-universal sets of size $n+1$ in$\mathbb{Z}[i]$, provided that $n$ is large enough. In a joint work with Jakub Byszewski and Mikolaj Fraczyk we extended their result to the rings of integers in any imaginary quadratic field. Petrov and Volkov also stated a conjecture about the minimial cardinality of $n$-universal sets in the ring of Gaussian integers. We give a strong counterexample to their conjecture by showing that in a ring of integers of any number field, for any natural $n$ there exists an $n$-universal set with $n+2$ elements. On the way, we discover a link with Euler-Kronecker constants.