  A popular method in combinatorial optimization is to express polytopes P,
which may potentially have exponentially many facets, as solutions of linear
programs that use few extra variables to reduce the number of constraints down
to a polynomial. After two decades of standstill, recent years have brought
amazing progress in showing lower bounds for the so called extension
complexity, which for a polytope P denotes the smallest number of inequalities
necessary to describe a higher dimensional polytope Q that can be linearly
projected on P.
  However, the central question in this field remained wide open: can the
perfect matching polytope be written as an LP with polynomially many
constraints?
  We answer this question negatively. In fact, the extension complexity of the
perfect matching polytope in a complete n-node graph is 2^Omega(n). By a known
reduction this also improves the lower bound on the extension complexity for
the TSP polytope from 2^Omega(n^1/2) to 2^Omega(n).
