Cardinality of a subset with a given property

Question

Suppose we are given two integers $n, q$ and a set $S := \{1,\ldots, q\}^n \times \{ 1,\ldots, n \}$.

I would like to establish the minimum natural number $k$ such that any subset $S' \subseteq S$ with $k$ elements satisfies the following property:

There exist two indexes $b_1,b_2 \in \{1,\ldots,n\}$ and two vectors $z,z' \in \{1,\ldots,q\}^n$ such that

1. $(z,b_1),(z,b_2),(z',b_1),(z',b_2) \in S'$;
2. $z_{b_1} = z'_{b_1}$ and $z_{b_2} \neq z'_{b_2}$ (where $z_i$ denotes the $i$-th entry of $z$).

Thanks in advance.