  The study of intersection problems in Extremal Combinatorics dates back
perhaps to 1938, when Paul Erd\H{o}s, Chao Ko and Richard Rado proved the
(first) `Erd\H{o}s-Ko-Rado theorem' on the maximum possible size of an
intersecting family of $k$-element subsets of a finite set. Since then, a
plethora of results of a similar flavour have been proved, for a range of
different mathematical structures, using a wide variety of different methods.
Structures studied in this context have included families of vector subspaces,
families of graphs, subsets of finite groups with given group actions, and of
course uniform hypergraphs with stronger or weaker intersection conditions
imposed. The methods used have included purely combinatorial ones such as
shifting/compressions, algebraic methods (including linear-algebraic, Fourier
analytic and representation-theoretic), and more recently, analytic,
probabilistic and regularity-type methods. As well as being natural problems in
their own right, intersection problems have connections with many other parts
of Combinatorics and with Theoretical Computer Science (and indeed with many
other parts of Mathematics), both through the results themselves, and the
methods used. In this survey paper, we discuss both old and new results (and
both old and new methods), in the field of intersection problems. Many
interesting open problems remain; we will discuss several. For expositional and
pedagogical purposes, we also take this opportunity to give slightly
streamlined versions of proofs (due to others) of several classical results in
the area. This survey is intended to be useful to PhD students, as well as to
more established researchers. It is a personal perspective on the field, and is
not intended to be exhaustive; we apologise for any omissions. It is an
expanded version of a paper that will appear in the Proceedings of the 29th
British Combinatorial Conference.
