A graph has a pair $(m,f)$ if it has an induced subgraph on $m$ vertices and $f$ edges. We write $(n,e)\rightarrow (m,f)$  if any graph on $n$ vertices and $e$ edges has a pair $(m,f)$.  Let  $$S(n,m,f)=\{e: ~(n,e)\rightarrow (m,f)\} ~{\rm and}$$     $$\sigma(m,f) =   \limsup_{n\rightarrow \infty}\frac{ |S(n,m,f)|}{\binom{n}{2}}.$$ These notions were first introduced and investigated by Erdős, Füredi, Rothschild, and Sós. They found five pairs $(m,f)$ with  $\sigma(m,f)=1$ and showed that for all other pairs $\sigma(m,f)\leq 2/3$.  We extend these results in two directions.

First, in a joint work with Weber, we show that not only $\sigma(m,f)$ can be zero, but also $S(n,m,f)$  could be empty for some pairs $(m,f)$ and any sufficiently large $n$. We call such pairs $(m,f)$ absolutely avoidable.

Second, we consider a natural analogue $\sigma_r(m,f)$ of $\sigma(m,f)$ in the setting of $r$-uniform hypergraphs.  Weber showed that for any $r\geq 3$ and  $m>r$,  $\sigma_r(m,f)=0$ for most values of $f$.  Surprisingly, it was not immediately clear whether there are nontrivial pairs $(m,f)$,  $(f\neq 0$, $f\neq \binom{m}{r}$,  $r\geq 3$),  for which $\sigma_r(m,f)>0$. In a joint work with Balogh, Clemen, and Weber we show that $\sigma_3(6,10)>0$ and conjecture that in the $3$-uniform case $(6,10)$ is the only such pair.

A Latin square of order n is an n by n grid filled with n different symbols so that every symbol occurs exactly once in each row and each column, while a transversal in a Latin square is a collection of cells which share no row, column or symbol. The Ryser-Brualdi-Stein conjecture states that every Latin square of order n should have a transversal with n-1 elements, and one with n elements if n is odd. In 2020, Keevash, Pokrovskiy, Sudakov and Yepremyan improved the long-standing best known bounds on this conjecture by showing that a transversal with n-O(log n/loglog n) elements exists in any Latin square of order n. In this talk, I will discuss how to show, for large n, that a transversal with n-1 elements always exists.

From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Will balloons reach the origin infinitely often or not? We answer this question for various underlying spaces. En route we find a new(ish) 0-1 law, and generalize bounds on independent sets that are factors of IID on trees. Joint work with Omer Angel and Gourab Ray.

In 1983 Thomassen conjectured that every graph of sufficiently large average degree has a subgraph of large girth and large average degree. While this conjecture remains open, recent evidence suggests that something stronger might be true; perhaps the subgraph can be made induced when a clique and biclique are forbidden. We overview our proof for removing 4-cycles from $K_{t,t}$-free bipartite graphs. Moreover, we discuss consequences to tau-boundedness, which is an analog of chi-boundedness.

A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$, the states (i.e. present or not present) of the edges in $S_1$ are independent of the states of the edges in $S_2$. Such models typically arise in renormalisation arguments applied to independent percolation models, or percolation models with finite range dependencies. A 1-independent model is said to percolate if the random subgraph has an infinite component with positive probability. In 2012 Balister and Bollobás defined $p_{\textrm{max}}(G)$ to be the supremum of those $p$ for which there exists a 1-independent bond percolation model on $G$ in which each edge is present in the random subgraph with probability at least $p$ but which does not percolate. A fundamental and challenging problem in this area is to determine, or give good bounds on, the value of $p_{\textrm{max}}(G)$ when $G$ is the lattice graph $\mathbb{Z}^2$. Since $p_{\textrm{max}}(\mathbb{Z}^n)\leq p_{\textrm{max}}(\mathbb{Z}^{n-1})$, it is also of interest to establish the value of $\lim_{n\to\infty}p_{\textrm{max}}(\mathbb{Z}^n)$.

In this talk we will present a significantly improved upper bound for this limit as well as improved upper and lower bounds for $p_{\textrm{max}}(\mathbb{Z}^2)$. We will also show that with high confidence we have $p_{\textrm{max}}(\mathbb{Z}^n)<p_{\textrm{max}}(\mathbb{Z}^2)$ for large $n$ and discuss some open problems concerning 1-independent models on other graphs.

This is joint work with Tom Johnston and Alex Scott.