Many important theorems and conjectures in combinatorics, such as the
theorem of Szemerédi on arithmetic progressions and the Erdős-Stone
Theorem in extremal graph theory, can be phrased as statements about
families of independent sets in certain uniform hypergraphs. In recent
years, an important trend in the area has been to extend such
classical results to the so-called `sparse random setting'. This line
of research has recently culminated in the breakthroughs of Conlon and
Gowers and of Schacht, who developed general tools for solving
problems of this type. Although these two papers solved very similar
sets of longstanding open problems, the methods used are very
different from one another and have different strengths and
weaknesses.
In this talk, we explain a third, completely different approach to
proving extremal and structural results in sparse random sets that
also yields their natural `counting' counterparts. We give a
structural characterization of the independent sets in a large class
of uniform hypergraphs.
Joint work with Robert Morris and Wojciech Samotij.