Abstract
Define a set of lines in
R
3
to be ``stacked'' with respect to
v∈
R
3
if, from a vantage point far away in the direction of
v
, the lines are linearly ordered by the ``crossing over'' relation. Given a collection of skew lines and a point
v
, we ask, what is the largest stacked subset that must be present among the lines? This question, which appears in a 2000 paper of Erdos, Hajnal and Pach, is intimately related to the well-known Erdos-Hajnal conjecture via the Milnor-Thom theorem. It was recently resolved by a powerful and very general theorem of Alon, Pach, Pinchasi, Radoicic, and Sharir. We describe these results and discuss several related issues, including a generalization to ``Erdos-Hajnal sets'' and an intriguing problem concerning the decomposability of semi-algebraic sets: Do all semi-algebraic sets belong to the set algebra generated by semigroups in
R
d
? Our main result is a resolution of this question in dimensions 1 and 2.