Geometric aspects of partially ordered sets
by
W. T. (Tom) Trotter
Arizona State University

This talk starts with problems and solutions readily understood by undergraduates. By the end, we will have moved on to unsolved problems where only the formulation of the problem remains clear. Solutions are welcomed!

An inclusion representation of a partially ordered set (poset) P=(X, P) is a family F = {S_x:x in X} so that x <= y in P if and only if S_x subset or equal S_y. Every poset has an inclusion representation, e.g., just take S_x={u:u <= x in P}, so it makes sense to place restrictions on the sets in the family. One very natural option is to require that the sets be geometric objects: rectangles, squares, circles, ellpses in the plane, or rectilinear solids, cubes, balls, or cones in R^t. Recall that the dimension of a poset P=(X, P) is the least t for which P is the intersection of t linear orders. It is easy to see that for each t >= 2, a poset has dimension at most t if and only if it has an inclusion representation using cubes (with sides parallel to the coordinate axes) in R^t-1. If we drop the restriction that the sides are parallel to the coordinate axes, then we only know that a poset of dimension 2t has an inclusion representation using cubes in R^t, and even the Alon/Scheinerman theory does not help us to settle whether this result is sharp when t >= 3. We also discuss the ramsey theoretic techniques used by Felsner, Fishburn and Trotter to show that there are finite three dimensional posets which do not have inclusion representations using circles in the plane.

Date received: April 18, 2001