Let $K_n$ denote the complete graph on $n$ vertices. Let us denote the vertices simply by $1,\ldots,n$. Suppose that, for each edge $ij$, with $1\leq i<j \leq n$, we assign an ordered basis $(p^+_{ij}, p^-_{ij})$ of $\mathbb{C}^2$, where we think of the latter as the space of complex polynomials of degree less than or equal to $1$.

Here is the proposed problem. An admissible choice is, for each oriented edge $ij$, as above, a choice of either $p^+_{ij}$ or $p^-_{ij}$. We then assign to the edge $ji$ the "other" choice (if you chose $p^+$ for $ij$, you must assign $p^-$ to $ji$, and vice versa).

Once an admissible choice is made, one then forms, for each $i$, the polynomial $p_i$, which is equal to the product of the chosen $p_{ij}$'s, for $j$ running over all values in $\{1,\ldots,n\}$, with $j \neq i$. Thus each $p_i$ is a polynomial of degree less than or equal to $n-1$.

Problem: given any initial assignment of ordered bases to the edges, does there always exist some admissible choice such that the corresponding polynomials $p_1,\ldots,p_n$ are linearly independent over $\mathbb{C}$?