Variations on the slope problem

The slope problem, in its most basic form, asks the question “Given n points in $R2$ which do not all lie on the same line, what is the smallest number of distinct slopes that they must determine?” The answer to this was settled in a 1980 paper by Peter Ungar, who used purely combinatorial methods to show that the answer was n if n was even, and n − 1 if n was odd.; This paper considers several generalizations of this problem, such as: (1) Since the combinatorial methods invoked by Ungar essentially prove something about oriented matroids, can we find a similar matroid result if we ignore the orientation? (2) What if the slope(s) between certain predetermined points are ignored? If we construct a graph which records which slopes we are interested in, do properties of the graph tell us anything about the number of slopes required? (3) Having considered the more general slope problem involving graphs, is there a signed graph analog of the slope problem?; As we will see, the answers to these questions are as follows. (1) There is a very strong analog of Ungar's result for matroids without orientation, and the corresponding smallest number of slopes turns out to grow much more slowly—more like $n$ instead of n. Furthermore, the question of when the known lower bound is attained is equivalent to a long standing problem in design theory concerning the existence of finite projective planes. (2) While the case of an arbitrary graph seems to be difficult, it turns out that nearly everything can be resolved in the case of threshold graphs. There are also some results for arbitrary graphs involving the chromatic number as well as Crapo's β-invariant. (3) There is certainly a type B analog, which is essentially the problem of point configurations in $R2$ which are centrally symmetric. Moreover, it turns out that there is a very nice analog of threshold graphs for type B, and again we can say nearly everything in this case.