We prove that there exist natural generalizations of the classical bootstrap percolation model on $\mathbb{Z}^2$ that have non-trivial critical probabilities, and moreover we characterize all homogeneous, local, monotone models with this property. Joint work with Paul Balister, Béla Bollobás and Paul Smith.

Let $P(M)$ be the matroid base polytope of a matroid $M$. A decomposition of $P(M)$ is a subdivision of the form $P(M)=\cup_{i=1}^t P(M_i)$ where each $P(M_i)$ is also a matroid base polytope for some matroid $M_i$, and for each $1\le i\neq j\le t$ the intersection $P(M_i)\cap P(M_j)$ is a face of both $P(M_i)$ and $P(M_j)$. In this talk, we shall discuss some results on hyperplane splits, that is, polytope decomposition when $t=2$. We present sufficient conditions for $M$ so $P(M)$ has a hyperplane split and a characterization when $P(M_i\oplus M_j)$ has a hyperplane split, where $M_i\oplus M_j$ denotes the direct sum of $M_i$ and $M_j$. We also show that $P(M)$ has not a hyperplane split if $M$ is binary. Finally, we present some recent results concerning the existence of decompositions with $t\ge 3$.

Given a tree $T$ with $n$ vertices, how large does $p$ need to be for it to be likely that a copy of $T$ appears in the binomial random graph $G(n,p)$? I will discuss this question, including recent work confirming a conjecture which gives a good answer to this question for trees with bounded maximum degree.