# Configurations from arrangement of lines

Suppose we have an arrangement of $n$ lines in the plane. This partitions the plane into regions. We can label each region by a string in $\{+1, -1\}^n$, where the $i$'th coordinate is +1 if the region lies above the $i$'th line and -1 if it lies below the $i$'th line. (By rotating the arrangement if necessary, we can assume that no line is perfectly vertical.)

Say that a subset $S$ of $\{+1,-1\}^n$ is realizable by a linear arrangement if there exists a linear arrangement such that the labels of the regions are exactly the strings in $S$. Is there previous study of which subsets of $\{+1, -1\}^n$ are realizable by linear arrangements? One can also ask the same question for hyperplane arrangements.

This problem is known as the question of realizability of pseudoline arrangements by straight lines. An arrangement of pseudolines is an arrangement of $n$ curves in the plane, such that any two curves intersect exactly once. For pseudolines, you can define the same labeling as the OP does. Pseudoline arrangements are related to oriented matroids, and you can use these to give a polynomial-time algorithm for telling whether a labeling of regions is realizable by a pseudoline arrangment.