Past Combinatorial Theory Seminar

We will discuss the mixing rate of the standard random walk on the giant

component of the random graph G(n,p). We tie down the mixing rate precisely

for all values of p greater than (1+c)/n for any positive constant c. We need

to develop a new bound on the mixing time of general Markov chains, inspired

by and extending work of Kannan and Lovasz. This is joint work with Nick

Fountoulakis.

We will examine how the various notions of partition regularity change as we change the ambient space. A typical question would be as follows. We say that the system of equations $Ax=b$, where $A$ is an integer matrix and $b$ is a (non-zero) integer vector, is partition regular if whenever the integers are finitely coloured there is a monochromatic vector $x$ with $Ax=b$. Rado proved that the system $Ax=b$ is partition regular if and only if it has a constant solution. What happens if the integers are replaced by the rationals, or the reals, or a more general ring? 

No previous knowledge of partition regularity is assumed. This is based on joint work with Paul Russell and joint work with Ben Barber, Neil Hindman and Dona Strauss.