Joint work with Robert Morris and Wojciech Samotij.
Many important theorems and conjectures in combinatorics, such as the
theorem of Szemer’edi on arithmetic progressions and the
Erd{\H{o}}s-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.