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The $r$-colour Ramsey number $R_r(k)$ is the minimum $n \in \mathbb{N}$ such that every $r$-colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove, for each fixed $r \geqslant 2$, that $$R_r(k) \leqslant e^{-\delta k} r^{rk}$$ for some constant $\delta = \delta(r) > 0$ and all sufficiently large $k \in \mathbb{N}$. For each $r \geqslant 3$, this is the first exponential improvement over the upper bound of Erd\H{o}s and Szekeres from 1935. In the case $r = 2$, it gives a different (and significantly shorter) proof of a recent result of Campos, Griffiths, Morris and Sahasrabudhe., Comment: 17 pages

The large combined field of view of the Geostationary Lightning Mapper (GLM) instruments onboard the GOES weather satellites makes them useful for studying the population of other atmospheric phenomena, such as bolides. Being a lightning mapper, GLM has many detection biases when applied to non-lightning and these systematics must be studied and properly accounted for before precise measurements of bolide flux can be ascertained. We developed a Bayesian Poisson regression model which simultaneously estimates instrumental biases and our statistic of principal interest: the latitudinal variation of bolide flux. We find that the estimated bias due to the angle of incident light upon the instrument corresponds roughly with the known sensitivity of the GLM instruments. We compare our latitudinal flux variation estimates to existing theoretical models and find our estimates consistent with GLM being strongly biased towards high-velocity bolides., Comment: To be published in Icarus vol. 408, 2024

The Ramsey number $R(k)$ is the minimum $n \in \mathbb{N}$ such that every red-blue colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove that \[ R(k) \leqslant (4 - \varepsilon)^k \] for some constant $\varepsilon > 0$. This is the first exponential improvement over the upper bound of Erd\H{o}s and Szekeres, proved in 1935., Comment: 57 pages, 8 figures

The set-colouring Ramsey number $R_{r,s}(k)$ is defined to be the minimum $n$ such that if each edge of the complete graph $K_n$ is assigned a set of $s$ colours from $\{1,\ldots,r\}$, then one of the colours contains a monochromatic clique of size $k$. The case $s = 1$ is the usual $r$-colour Ramsey number, and the case $s = r - 1$ was studied by Erd\H{o}s, Hajnal and Rado in 1965, and by Erd\H{o}s and Szemer\'edi in 1972. The first significant results for general $s$ were obtained only recently, by Conlon, Fox, He, Mubayi, Suk and Verstra\"ete, who showed that $R_{r,s}(k) = 2^{\Theta(kr)}$ if $s/r$ is bounded away from $0$ and $1$. In the range $s = r - o(r)$, however, their upper and lower bounds diverge significantly. In this note we introduce a new (random) colouring, and use it to determine $R_{r,s}(k)$ up to polylogarithmic factors in the exponent for essentially all $r$, $s$ and $k$., Comment: 12 pages, submitted version, added Conjecture 5.1

A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of these objects was initiated by Erd\H{o}s in 1950, and over the following decades he asked many questions about them. Most famously, he asked whether there exist covering systems with distinct moduli whose minimum modulus is arbitrarily large. This problem was resolved in 2015 by Hough, who showed that in any such system the minimum modulus is at most $10^{16}$. The purpose of this note is to give a gentle exposition of a simpler and stronger variant of Hough's method, which was recently used to answer several other questions about covering systems. We hope that this technique, which we call the distortion method, will have many further applications in other combinatorial settings., Comment: 8 pages, expository note

CheckSync provides applications with high availability via runtime-integrated checkpointing. This allows CheckSync to take checkpoints of a process running in a memory-managed language (Go, for now), which can be resumed on another machine after a failure. CheckSync uses the runtime to checkpoint only the process' live memory, doing without requiring significant changes to applications. CheckSync maintains the ease of use provided by virtual machines for the applications it supports without requiring that an entire virtual machine image be snapshotted. Because CheckSync captures only the memory used by an application, it produces checkpoints that are smaller (by an order of magnitude) than virtual machine snapshots if the memory footprint of the application is relatively small compared to the state of the rest of the operating system. Additionally, when running go-cache, a popular in-memory key/value store, CheckSync reduces throughput by only 12% compared to the 78% throughput loss when using go-cache's snapshot functionality, the 45% loss when using CRIU, and the 68% loss when using virtual machine live migration., Comment: 14 pages, 6 figures

In 1957, Hadwiger conjectured that every convex body in $\mathbb{R}^d$ can be covered by $2^d$ translates of its interior. For over 60 years, the best known bound was of the form $O(4^d \sqrt{d} \log d)$, but this was recently improved by a factor of $e^{\Omega(\sqrt{d})}$ by Huang, Slomka, Tkocz and Vritsiou. In this note we take another step towards Hadwiger's conjecture by deducing an almost-exponential improvement from the recent breakthrough work of Chen, Klartag and Lehec on Bourgain's slicing problem. More precisely, we prove that, for any convex body $K \subset \mathbb{R}^d$, $$\exp\bigg( - \Omega\bigg( \frac{d}{(\log d)^8} \bigg) \bigg) \cdot 4^d$$ translates of $\text{int}(K)$ suffice to cover $K$. We also show that a positive answer to Bourgain's slicing problem would imply an exponential improvement for Hadwiger's conjecture., Comment: 10 pages

In many interacting particle systems, relaxation to equilibrium is thought to occur via the growth of 'droplets', and it is a question of fundamental importance to determine the critical length at which such droplets appear. In this paper we construct a mechanism for the growth of droplets in an arbitrary finite-range monotone cellular automaton on a $d$-dimensional lattice. Our main application is an upper bound on the critical probability for percolation that is sharp up to a constant factor in the exponent. Our method also provides several crucial tools that we expect to have applications to other interacting particle systems, such as kinetically constrained spin models on $\mathbb{Z}^d$. This is one of three papers that together confirm the Universality Conjecture of Bollob\'as, Duminil-Copin, Morris and Smith., Comment: 159 pages, 4 figures, numerous minor improvements based on a very helpful referee report

In this paper we study monotone cellular automata in $d$ dimensions. We develop a general method for bounding the growth of the infected set when the initial configuration is chosen randomly, and then use this method to prove a lower bound on the critical probability for percolation that is sharp up to a constant factor in the exponent for every 'critical' model. This is one of three papers that together confirm the Universality Conjecture of Bollob\'as, Duminil-Copin, Morris and Smith., Comment: 65 pages, 2 figures, minor update, expanded outline of the proof

We study monotone cellular automata (also known as $\mathcal{U}$-bootstrap percolation) in $\mathbb{Z}^d$ with random initial configurations. Confirming a conjecture of Balister, Bollob\'as, Przykucki and Smith, who proved the corresponding result in two dimensions, we show that the critical probability is non-zero for all subcritical models., Comment: 23 pages, 2 figures, added Theorem 7.1

Ultra-short-period (USP) exoplanets have orbital periods shorter than one day. Precise masses and radii of USPs could provide constraints on their unknown formation and evolution processes. We report the detection and characterization of the USP planet GJ 367b using high precision photometry and radial velocity observations. GJ 367b orbits a bright (V-band magnitude = 10.2), nearby, red (M-type) dwarf star every 7.7 hours. GJ 367b has a radius of $0.718 \pm 0.054$ Earth-radii, a mass of $0.546 \pm 0.078$ Earth-masses, making it a sub-Earth. The corresponding bulk density is $8.106 \pm 2.165$ g cm$^-3$, close to that of iron. An interior structure model predicts the planet has an iron core radius fraction of $86 \pm 5\%$, similar to Mercury's interior., Comment: Note: "This is the author's version of the work. It is posted here by permission of the AAAS for personal use, not for redistribution. The definitive version was published in Science , (2021-12-03), doi: 10.1126/science.aay3253"