Suppose $S$ is a k-ary relation on $\mathbb{N}$. Given $1 \leq i \leq k$, define $\pi_i(S)$ in the following way: $$\left \{s_i | (s_1, s_2, \cdots s_i \cdots s_k) \in S \right \}$$. We call $S$ nice if for any component $i$, either the projection $\pi_i(S)$ or its complement is finite. I want...