Variations on the Slope Problem

The slope problem, in its most basic form, asks the question “Given n points in R 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 (i.e. of type A), is there any B analog to 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. 2. While the case of an arbitrary graph seems to be difficult, it appears that much can be said in the case of chordal graphs (and even more so for the class of threshold graphs, which are a subclass of chordal graphs). There are some initial general results, however, involving the chromatic number of a graph, as well as Crapo’s β-invariant. 3. There is certainly a type B analog, which is essentially the problem of point configurations in R which are centrally symmetric.