  The disjointness problem - where Alice and Bob are given two subsets of $\{1,
\dots, n\}$ and they have to check if their sets intersect - is a central
problem in the world of communication complexity. While both deterministic and
randomized communication complexities for this problem are known to be
$\Theta(n)$, it is also known that if the sets are assumed to be drawn from
some restricted set systems then the communication complexity can be much
lower. In this work, we explore how communication complexity measures change
with respect to the complexity of the underlying set system. The complexity
measure for the set system that we use in this work is the Vapnik-Chervonenkis
(VC) dimension. More precisely, on any set system with VC dimension bounded by
$d$, we analyze how large can the deterministic and randomized communication
complexities be, as a function of $d$ and $n$.
  In this paper, we construct two natural set systems of VC dimension $d$,
motivated from geometry. Using these set systems we show that the deterministic
and randomized communication complexity can be $\widetilde{\Theta}\left(d\log
\left( n/d \right)\right)$ for set systems of VC dimension $d$ and this matches
the deterministic upper bound for all set systems of VC dimension $d$. We also
study the deterministic and randomized communication complexities of the set
intersection problem when sets belong to a set system of bounded VC dimension.
We show that there exists set systems of VC dimension $d$ such that both
deterministic and randomized (one-way and multi-round) complexity for the set
intersection problem can be as high as $\Theta\left( d\log \left( n/d \right)
\right)$, and this is tight among all set systems of VC dimension $d$.
