arXiv subjects

Patrick Morris

Publications and source records attributed to Patrick Morris.

32 records · Page 2Linked to original sources

arXiv2026

Monochromatic products in random integer sets

A well-known consequence of Schur's theorem is that for $r\in \mathbb{N}$, if $n$ is sufficiently large, then any $r$-colouring of $[n]$ results in monochromatic $a,b,c\in [n]$ such that $ab=c$. In this paper we are interested in the threshold at which the binomial random set $[n]_p$ almost surely inherits this Ramsey-type property. In particular for $r=2$ colours, we show that this threshold lies between $n^{-1/9-o(1)}$ and $n^{-1/11}$. Whilst analogous questions for solutions to (sets of) linear equations are now well understood, our work suggests that both the behaviour of the thresholds and the proof methods needed to determine them differ substantially in the non-linear setting.

math.CO↗

arXiv2025

Slow graph bootstrap percolation III: Chain constructions

For graphs $H$, we study the extremal function $M_H(n)$ which is the maximum running time (until stabilisation) of an $H$-bootstrap percolation process on $n$ vertices. Building on previous work in the clique case $H=K_k$, we develop a general framework of chain constructions. We demonstrate the flexibility of this framework by applying several variations of the method to give lower bounds on $M_H(n)$ for a wide variety of different graphs $H$ including dense graphs, random graphs and complete bipartite graphs. In particular, we focus on the question of whether $M_H(n)$ is (almost) quadratic or not and our lower bounds develop connections with additive combinatorics, utilising constructions of sets free of solutions to certain linear equations. Finally, our lower bounds are complemented by upper bounds which connect $M_H(n)$ to other problems in extremal graph theory such as the Ruzsa-Szemerédi (6,3)-Theorem.

math.CO↗

arXiv2024

Slow graph bootstrap percolation II: Accelerating properties

For a graph $H$ and an $n$-vertex graph $G$, the $H$-bootstrap process on $G$ is the process which starts with $G$ and, at every time step, adds any missing edges on the vertices of $G$ that complete a copy of $H$. This process eventually stabilises and we are interested in the extremal question raised by Bollobás of determining the maximum running time (number of time steps before stabilising) of this process over all possible choices of $n$-vertex graph $G$. In this paper, we initiate a systematic study of the asymptotics of this parameter, denoted $M_H(n)$, and its dependence on properties of the graph $H$. Our focus is on $H$ which define relatively fast bootstrap processes, that is, with $M_H(n)$ being at most linear in $n$. We study the graph class of trees, showing that one can bound $M_T(n)$ by a quadratic function in $v(T)$ for all trees $T$ and all $n$. We then go on to explore the relationship between the running time of the $H$-process and the minimum vertex degree and connectivity of $H$.

math.CO↗

arXiv2024

Speeding up random walk mixing by starting from a uniform vertex

The theory of rapid mixing random walks plays a fundamental role in the study of modern randomised algorithms. Usually, the mixing time is measured with respect to the worst initial position. It is well known that the presence of bottlenecks in a graph hampers mixing and, in particular, starting inside a small bottleneck significantly slows down the diffusion of the walk in the first steps of the process. The average mixing time is defined to be the mixing time starting at a uniformly random vertex and hence is not sensitive to the slow diffusion caused by these bottlenecks. In this paper we provide a general framework to show logarithmic average mixing time for random walks on graphs with small bottlenecks. The framework is especially effective on certain families of random graphs with heterogeneous properties. We demonstrate its applicability on two random models for which the mixing time was known to be of order $(\log n)^2$, speeding up the mixing to order $\log n$. First, in the context of smoothed analysis on connected graphs, we show logarithmic average mixing time for randomly perturbed graphs of bounded degeneracy. A particular instance is the Newman-Watts small-world model. Second, we show logarithmic average mixing time for supercritically percolated expander graphs. When the host graph is complete, this application gives an alternative proof that the average mixing time of the giant component in the supercritical Erdős-Rényi graph is logarithmic.

math.PR↗

arXiv2023

Eta Carinae: the dissipating occulter is an extended structure

Previous STIS long-slit observations of Eta Carinae identified numerous absorption features in both the stellar spectrum, and in the adjacent nebular spectra, along our line-of-sight. The absorption features became temporarily stronger when the ionizing FUV radiation field was reduced by the periastron passage of the secondary star. Subsequently, dissipation of a dusty structure in our LOS has led to a long-term increase in the apparent magnitude of \ec, an increase in the ionizing UV radiation, and the disappearance of absorptions from multiple velocity-separated shells extending across the foreground Homunculus lobe. We use HST/STIS spectro-images, coupled with published infrared and radio observations, to locate this intervening dusty structure. Velocity and spatial information indicate the occulter is ~1000 au in front of Eta Carinae. The Homunculus is a transient structure composed of dusty, partially-ionized ejecta that eventually will disappear due to the relentless rain of ionizing radiation and wind from the current binary system along with dissipation and mixing with the ISM. This evolving complex continues to provide an astrophysical laboratory that changes on human timescales.

astro-ph.SR↗

arXiv2023

Clique factors in pseudorandom graphs

An $n$-vertex graph is said to to be $(p,β)$-bijumbled if for any vertex sets $A,B\subseteq V(G)$, we have \[e(A,B)=p|A||B|\pm β\sqrt{|A||B|}.\] We prove that for any $3\leq r\in \mathbb{N}$ and $c>0$ there exists an $\varepsilon>0$ such that any $n$-vertex $(p,β)$-bijumbled graph with $n\in r \mathbb{N}$, $δ(G)\geq cpn$ and $β\leq \varepsilon p^{r-1}n$, contains a $K_r$-factor. This implies a corresponding result for the stronger pseudorandom notion of $(n,d,λ)$-graphs. For the case of triangle factors, that is when $r=3$, this result resolves a conjecture of Krivelevich, Sudakov and Szabó from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result and a result of Han, Kohayakawa, Person and the author, we can conclude that the same condition of $β=o(p^2n)$ actually guarantees that a $(p,β)$-bijumbled graph $G$ contains every graph on $n$ vertices with maximum degree at most 2.

math.CO↗

arXiv2022

The orbital kinematics of eta Carinae over three periastra with a possible detection of the elusive secondary's motion

The binary eta Carinae is the closest example of a very massive star, which may have formed through a merger during its Great Eruption in the mid-nineteenth century. We aimed to confirm and improve the kinematics using a spectroscopic data set taken with the CTIO 1.5 m telescope over the time period of 2008-2020, covering three periastron passages of the highly eccentric orbit. We measure line variability of H-alpha and H-beta, where the radial velocity and orbital kinematics of the primary star were measured from the H-beta emission line using a bisector method. At phases away from periastron, we observed the He II 4686 emission moving opposite the primary star, consistent with a possible Wolf-Rayet companion, although with a seemingly narrow emission line. This could represent the first detection of emission from the companion.

astro-ph.SR↗

arXiv2022

Eta Carinae: an evolving view of the central binary, its interacting winds and its foreground ejecta

FUV spectra of Eta Car, recorded across two decades with HST/STIS, document multiple changes in resonant lines caused by dissipating extinction in our line of sight. The FUV flux has increased nearly ten-fold which has led to increased ionization of the multiple shells within the Homunculus and photo-destruction of molecular hydrogen. Comparison of observed resonant line profiles with CMFGEN model profiles allows separation of wind-wind collision and shell absorptions from the primary wind, P Cygni profiles.The dissipating occulter preferentially obscured the central binary and interacting winds relative to the very extended primary wind. We are now able to monitor changes in the colliding winds with orbital phase. High velocity transient absorptions occurred across the most recent periastron passage, indicating acceleration of the primary wind by the secondary wind which leads to a downstream, high velocity bowshock that is newly generated every orbital period. There is no evidence of changes in the properties of the binary winds.

astro-ph.SR↗

arXiv2021

Factors and loose Hamilton cycles in sparse pseudo-random hypergraphs

We investigate the emergence of spanning structures in sparse pseudo-random $k$-uniform hypergraphs, using the following comparatively weak notion of pseudo-randomness. A $k$-uniform hypergraph $H$ on $n$ vertices is called $(p,α,ε)$-pseudo-random if for all (not necessarily disjoint) vertex subsets $A_1,\dots, A_k{\subseteq} V(H)$ with $|A_1|\cdots |A_k|{\geq}αn^{k}$ we have $$e(A_1,\dots, A_k)=(1\pmε)p |A_1|\cdots |A_k|.$$ For any linear $k$-uniform $F$ we provide a bound on $α=α(n)$ in terms of $p=p(n)$ and $F$, such that (under natural divisibility assumptions on $n$) any $k$-uniform $\big(p,α, o(1)\big)$-pseudo-random $n$-vertex hypergraph $H$ with a mild minimum vertex degree condition contains an $F$-factor. The approach also enables us to establish the existence of loose Hamilton cycles in sufficiently pseudo-random hypergraphs and all results imply corresponding bounds for stronger notions of hypergraph pseudo-randomness such as jumbledness or large spectral gap. As a consequence, $\big(p,α, o(1)\big)$-pseudo-random $k$-graphs as above contain: $(i)$ a perfect matching if $α=o(p^{k})$ and $(ii)$ a loose Hamilton cycle if $α=o(p^{k-1})$. This extends the works of Lenz--Mubayi, and Lenz--Mubayi--Mycroft who studied the analogous problems in the dense setting.

math.CO↗

arXiv2021

Factors in randomly perturbed hypergraphs

We determine, up to a multiplicative constant, the optimal number of random edges that need to be added to a $k$-graph $H$ with minimum vertex degree $Ω(n^{k-1})$ to ensure an $F$-factor with high probability, for any $F$ that belongs to a certain class $\mathcal{F}$ of $k$-graphs, which includes, e.g., all $k$-partite $k$-graphs, $K_4^{(3)-}$ and the Fano plane. In particular, taking $F$ to be a single edge, this settles a problem of Krivelevich, Kwan and Sudakov [Combin. Probab. Comput. 25 (2016), 909--927]. We also address the case in which the host graph $H$ is not dense, indicating that starting from certain such $H$ is essentially the same as starting from an empty graph (namely, the purely random model).

math.CO↗

arXiv2020

Ryser's Conjecture for $t$-intersecting hypergraphs

A well-known conjecture, often attributed to Ryser, states that the cover number of an $r$-partite $r$-uniform hypergraph is at most $r - 1$ times larger than its matching number. Despite considerable effort, particularly in the intersecting case, this conjecture remains wide open, motivating the pursuit of variants of the original conjecture. Recently, Bustamante and Stein and, independently, Király and Tóthmérész considered the problem under the assumption that the hypergraph is $t$-intersecting, conjecturing that the cover number $τ(\mathcal{H})$ of such a hypergraph $\mathcal{H}$ is at most $r - t$. In these papers, it was proven that the conjecture is true for $r \leq 4t-1$, but also that it need not be sharp; when $r = 5$ and $t = 2$, one has $τ(\mathcal{H}) \leq 2$. We extend these results in two directions. First, for all $t \geq 2$ and $r \leq 3t-1$, we prove a tight upper bound on the cover number of these hypergraphs, showing that they in fact satisfy $τ(\mathcal{H}) \leq \lfloor(r - t)/2 \rfloor + 1$. Second, we extend the range of $t$ for which the conjecture is known to be true, showing that it holds for all $r \leq \frac{36}{7}t-5$. We also introduce several related variations on this theme. As a consequence of our tight bounds, we resolve the problem for $k$-wise $t$-intersecting hypergraphs, for all $k \geq 3$ and $t \geq 1$. We further give bounds on the cover numbers of strictly $t$-intersecting hypergraphs and the $s$-cover numbers of $t$-intersecting hypergraphs.

math.CO↗

arXiv2020

Tilings in randomly perturbed graphs: bridging the gap between Hajnal-Szemerédi and Johansson-Kahn-Vu

A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint copies of $K_r$ that together cover all the vertices in $G$. In this paper we consider perfect $K_r$-tilings in the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze and Martin where one starts with a dense graph and then adds $m$ random edges to it. Specifically, given any fixed $0< α<1-1/r$ we determine how many random edges one must add to an $n$-vertex graph $G$ of minimum degree $δ(G) \geq αn$ to ensure that, asymptotically almost surely, the resulting graph contains a perfect $K_r$-tiling. As one increases $α$ we demonstrate that the number of random edges required `jumps' at regular intervals, and within these intervals our result is best-possible. This work therefore closes the gap between the seminal work of Johansson, Kahn and Vu (which resolves the purely random case, i.e., $α=0$) and that of Hajnal and Szemerédi (which demonstrates that for $α\geq 1-1/r$ the initial graph already houses the desired perfect $K_r$-tiling).

math.CO↗

arXiv2008

Properties of protostars in the Elephant Trunk globule IC 1396A

Extremely red objects, identified in the early Spitzer Space Telescope observations of the bright-rimmed globule IC 1396A and photometrically classified as Class I protostars Class II T Tauri stars based on their mid-infrared colors, were observed spectroscopically at 5.5 to 38 microns (IRS), at the 22 GHz water maser frequency (GBT), and in the optical (Palomar). The sources photometrically identified as Class I are confirmed as objects dominated by accretion luminosity from dense envelopes, with accretion rates 1e-5 to 1e-6 Msun/yr. The ice/silicate absorption ratio in the envelope is exceptionally low for the IC 1396A protostars, compared to those in nearby star-forming regions, suggesting the envelope chemistry is altered by the radiation field or globule pressure. Only one 22 GHz water maser was detected in IC 1396A; its infrared counterpart has luminosity <0.1 Lsun, the first H2O maser from such a low-luminosity object. The objects photometrically classified as Class II are confirmed as classical T Tauri stars with warm, luminous disks. The disk properties change significantly with source luminosity: low-mass (G-K) stars have prominent 9-11 micron emission features due to amorphous silicates while higher-mass (A-F) stars have weaker features requiring abundant crystalline silicates. The distribution of Class I sources is concentrated within the molecular globule, while the Class II sources are more widely scattered. Combined with the spectral results, this suggests two phases of star formation, the first (4 Myr ago) leading to the widespread Class II sources and the central O star of IC 1396, and the second (<1 Myr ago) occurring within the globule.

astro-ph↗

arXiv2003

The Mid-Infrared Spectrum of the Zodiacal and Exozodiacal Light

The zodiacal light is the dominant source of the mid-infrared sky brightness seen from Earth, and exozodiacal light is the dominant emission from planetary and debris systems around other stars. We observed the zodiacal light spectrum with ISOCAM over 5-16 over a wide range of orientations relative to the Sun and the ecliptic. We present theoretical models for a wide range of particle size distributions and compositions. The observed temperature is as expected for large (>10 um radius), low-albedo (< 0.08), rapidly-rotating, grey particles 1 AU from the Sun. In addition to the continuum, we detect a weak excess in the 9-11 um range, with an amplitude of 6% of the continuum. The shape of the feature can be matched by a mixture of silicates: amorphous forsterite/olivine, dirty crystalline olivine, and a hydrous silicate (montmorillonite). The presence of hydrous silicate suggests the parent bodies of those particles were formed in the inner solar nebula. Large particles dominate the size distribution, but at least some small particles (radii ~1 um) are required to produce the silicate emission feature. To compare the properties of zodiacal dust to dust around other main sequence stars, we reanalyzed the exozodiacal light spectrum for Beta Pic. The exozodiacal spectra are dominated by cold dust, with emission peaking in the far-infrared, while the zodiacal spectrum peaks around 20 um. The shape of the silicate feature from Beta Pic is nearly identical to that derived from the ISO spectrum of 51 Oph; both exozodiacal features are very different from that of the zodiacal light. The exozodiacal features are roughly triangular, peaking at 10.3 um while the zodiacal feature is more boxy.

astro-ph↗