The discrete cube {0,1}d is a fundamental combinatorial structure. A
subcube of {0,1}d is a subset of 2k of its points formed by fixing k
coordinates and allowing the remaining d−k to vary freely. The subcube
structure of the discrete cube is surprisingly complicated and there are many
open questions relating to it.
This paper is concerned with patterns of intersections among subcubes of the
discrete cube. Two sample questions along these lines are as follows: given a
family of subcubes in which no r+1 of them have non-empty intersection, how
many pairwise intersections can we have? How many subcubes can we have if among
them there are no k which have non-empty intersection and no l which are
pairwise disjoint? These questions are naturally expressed as Tur\'an and
Ramsey type questions in intersection graphs of subcubes where the intersection
graph of a family of sets has one vertex for each set in the family with two
vertices being adjacent if the corresponding subsets intersect.
Tur\'an and Ramsey type problems are at the heart of extremal combinatorics
and so these problems are mathematically natural. However, a second motivation
is a connection with some questions in social choice theory arising from a
simple model of agreement in a society. Specifically, if we have to make a
binary choice on each of n separate issues then it is reasonable to assume
that the set of choices which are acceptable to an individual will be
represented by a subcube. Consequently, the pattern of intersections within a
family of subcubes will have implications for the level of agreement within a
society.
We pose a number of questions and conjectures relating directly to the
Tur\'an and Ramsey problems as well as raising some further directions for
study of subcube intersection graphs.Comment: 18 page