We present a short and simple proof of the celebrated hypergraph container theorem of Balogh–Morris–Samotij and Saxton–Thomason. On a high level, our argument utilises the idea of iteratively taking vertices of largest degree from an independent set and constructing a hypergraph of lower uniformity which preserves independent sets and inherits edge distribution. The original algorithms for constructing containers also remove in each step vertices of high degree, which are not in the independent set. Our modified algorithm postpones this until the end, which surprisingly results in a significantly simplified analysis.

Counting independent sets in graphs and hypergraphs under a variety of restrictions is a classical question with a long history. It is the subject of the celebrated container method which found numerous spectacular applications over the years. We consider the question of how many independent sets we can have in a graph under structural restrictions. We show that any $n$-vertex graph with independence number $\alpha$ without $bK_a$ as an induced subgraph has at most $n^{O(1)} \cdot \alpha ^{O(\alpha )}$ independent sets. This substantially improves the trivial upper bound of $n^{\alpha },$ whenever $\alpha \le n^{o(1)}$ and gives a characterisation of graphs forbidding which allows for such an improvement. It is also in general tight up to a constant in the exponent since there exist triangle-free graphs with $\alpha ^{\Omega (\alpha )}$ independent sets. We also prove that if one in addition assumes the ground graph is chi-bounded one can improve the bound to $n^{O(1)} \cdot 2^{O(\alpha )}$ which is tight up to a constant factor in the exponent.

We prove a Poisson process approximation result for stabilising functionals of a determinantal point process. Our results use concrete couplings of determinantal processes with different Palm measures and exploit their association properties. Second, we focus on the Ginibre process and show in the asymptotic scenario of an increasing observation window that the process of points with a large nearest neighbour distance converges after a suitable scaling to a Poisson point process. As a corollary, we obtain the scaling of the maximum nearest neighbour distance in the Ginibre process, which turns out to be different from its analogue for independent points.

A graph $H$ is said to be common if the number of monochromatic labelled copies of $H$ in a red/blue edge colouring of a large complete graph is asymptotically minimised by a random colouring in which each edge is equally likely to be red or blue. We extend this notion to an off-diagonal setting. That is, we define a pair $(H_1,H_2)$ of graphs to be $(p,1-p)$-common if a particular linear combination of the density of $H_1$ in red and $H_2$ in blue is asymptotically minimised by a random colouring in which each edge is coloured red with probability $p$ and blue with probability $1-p$. Our results include off-diagonal extensions of several standard theorems on common graphs and novel results for common pairs of graphs with no natural analogue in the classical setting.

A random temporal graph is an Erdős-Rényi random graph $G(n,p)$, together with a random ordering of its edges. A path in the graph is called increasing if the edges on the path appear in increasing order. A set $S$ of vertices forms a temporal clique if for all $u,v \in S$, there is an increasing path from $u$ to $v$. Becker, Casteigts, Crescenzi, Kodric, Renken, Raskin and Zamaraev [(2023) Giant components in random temporal graphs. arXiv,2205.14888] proved that if $p=c\log n/n$ for $c\gt 1$, then, with high probability, there is a temporal clique of size $n-o(n)$. On the other hand, for $c\lt 1$, with high probability, the largest temporal clique is of size $o(n)$. In this note, we improve the latter bound by showing that, for $c\lt 1$, the largest temporal clique is of constant size with high probability.

We initiate a study of large deviations for block model random graphs in the dense regime. Following [14], we establish an LDP for dense block models, viewed as random graphons. As an application of our result, we study upper tail large deviations for homomorphism densities of regular graphs. We identify the existence of a ‘symmetric’ phase, where the graph, conditioned on the rare event, looks like a block model with the same block sizes as the generating graphon. In specific examples, we also identify the existence of a ‘symmetry breaking’ regime, where the conditional structure is not a block model with compatible dimensions. This identifies a ‘reentrant phase transition’ phenomenon for this problem – analogous to one established for Erdős–Rényi random graphs [13, 14]. Finally, extending the analysis of [34], we identify the precise boundary between the symmetry and symmetry breaking regimes for homomorphism densities of regular graphs and the operator norm on Erdős–Rényi bipartite graphs.

We show that on every affine Weyl group natural random walks are noise sensitive in total variation.