School of Mathematical Sciences

Department Colloquium

Schreiber 006, 12:15

University of Cambridge and Tel Aviv University

Many important theorems and conjectures in combinatorics, such as the theorem

Abstract:

of Szemeredi on arithmetic progressions and the Erdos-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 extremal and probabilistic combinatorics 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 approaches solved very similar

sets of longstanding open problems, the methods used are very different from one another and

have different strengths and weaknesses.

In the talk, we describe 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 by

showing that every independent set is almost contained in one of a small number of relatively

sparse sets. We then show how to derive many interesting results as fairly straightforward

consequences of this abstract theorem.

Based on joint work with Jozsef Balogh and Robert Morris.

Coffee will be served at 12:00 before the lecture

at Schreiber building 006