TITLE: The Distinct Volumes Problem
AUTHORS:
David Conlon, Jacob Fox, William Gasarch (the speaker),
David Harris, Douglas Ulrich, Sam Zbarsky.
ABSTRACT:
Given n points in the plane, we want a LARGE subset so that every pair of points has a DIFFERENT distance. We show that there is such a set of size roughly n^{1/3}. If the points are in R^d we can get roughly n^{1/3d-3} (We can do slightly better but these are ther results we'll present.)
Given n points in the plane, no three colinear, we want a LARGE subset so that every triple of points has a DIFFERENT area. We show that there is such a set of size roughly n^{1/10}. If the points are in R^d we can get roughly n^{1/5d}. (We can do slightly better but these are ther results we'll present.)
Given n points in 3-dim space, no four in the same plane, we want a LARGE subset so that every foursome of points has a DIFFERENT volume. We show that there is such a set of size roughly n^{1/14} If the points are in R^d we can get roughly n^{1/7d}. (We can do slightly better but these are ther results we'll present.)
In addition to discussing the mathematics we will discuss how one comes up with problems to work on, finds co-authors, gets papers written and published, using this papers story as an example.