arXiv subjects

Patrick Morris

Publications and source records attributed to Patrick Morris.

At least 19 recordsLinked to original sources

arXiv2026

Counting subsets of integers free of arithmetic configurations

Cameron and Erd\H{o}s asked if the number of sets free of arithmetic progressions of length $k$ is $2^{r_k(n)(1+o(1))}$, where $r_k(n)$ is the maximum cardinality of a $k$-AP-free subset of $\{1, \dots, n\}$. Balogh, Liu and Sharifzadeh made significant progress on this question showing that it is $2^{O(r_k(n))}$ for an infinite sequence of $n$. We improve their result in two ways. On the one hand, we prove that, for $k\geq 5$, the number of $k$-AP-free sets in $[n]$ is $2^{r_k(n)(1+o(1))}$ for an infinite sequence of $n$, solving the question of Cameron and Erd\H{o}s for infinitely many values. On the other hand, we also prove that for $k \geq 3$ and all $n$ the number of $k$-AP-free sets in $[n]$ is $2^{O(r_k(n))}$. These results are in fact special cases of a general framework that we develop to count families of sets excluding certain arithmetic patterns, which applies as long as the corresponding extremal threshold satisfies certain Behrend-type lower bounds. As further examples, we get analogous results for solution sets to almost all systems of linear equations as well as counting versions of the multidimensional Szemer\'edi theorem.

math.CO↗

arXiv2026

Universal probability bounds for partial Latin squares

This paper studies the probability of substructures occurring in random Latin squares. Our main result states that if $\alpha,\beta>0$ are such that $2\alpha+\beta<1$, then there are positive constants $\delta = \delta(\alpha, \beta)$ and $\Delta = \Delta(\alpha, \beta)$ such that if $P$ is a partial Latin square of order $n$ with $k = k(n)$ non-empty cells occupying at most $\alpha n$ rows and $\beta n$ columns, the probability that a random Latin square of order $n$ contains $P$ lies between $(\delta/n)^k$ and $(\Delta/n)^k$. We apply this result to subsquares in random Latin squares to obtain the first proof of the fact that the expected number of subsquares of order $3$ in a random Latin square of order $n$ is non-vanishing as $n \to \infty$. We are also able to provide the best known asymptotics for the expected number of subsquares of order $a$ in a random Latin square of order $n$ when $2<a=o(n^{1/2})$. Finally, we discuss the implications of our result on other configurations in random Latin squares as well as on completions of partial Latin squares.

math.CO↗

arXiv2026

Graph bootstrap percolation -- a discovery of slowness

Graph bootstrap percolation is a discrete-time process capturing the spread of a virus on the edges of $K_n$. Given an initial set $G\subseteq K_n$ of infected edges, the transmission of the virus is governed by a fixed graph $H$: in each round of the process any edge $e$ of $K_n$ that is the last uninfected edge in a copy of $H$ in $K_n$ gets infected as well. Once infected, edges remain infected forever. The process was introduced by Bollob\'as in 1968 in the context of weak saturation and has since inspired a vast array of beautiful mathematics. The main focus of this survey is the extremal question of how long the infection process can last before stabilising. We give an exposition of our recent systematic study of this maximum running time and the influence of the infection rule $H$. The topic turns out to possess a wide variety of interesting behaviour, with connections to additive, extremal and probabilistic combinatorics. Along the way we encounter a number of surprises and attractive open problems.

math.CO↗

arXiv2026

A note on multicolour Ramsey numbers and random sphere graphs

The Ramsey number $r(t;\ell)$ is the smallest $n$ such that every $\ell$-coloring of the edges of $K_n$ gives a monochromatic $K_{t}$. In recent years, there have been several improvements on asymptotic lower bounds for these numbers when $\ell\geq 3$. This started with a breakthrough result of Conlon and Ferber, followed by further improvements of Wigderson and then Sawin. Building on the previous approaches, Sawin used blowups of an unbalanced binomial random graph to show that there is some explicit constant $\delta_*\approx 0.383796$ such that $r(t;\ell)\geq 2^{\delta_*(\ell-2)t+t/2+o(t)}$. In this short note, we show that one can get an exponential improvement in this bound by replacing the use of a binomial random graph with a random sphere graph, a model which which has recently been applied by Ma, Shen and Xie in a breakthrough on lower bounds for (2-colour) Ramsey numbers in the (slightly) off-diagonal setting.

math.CO↗

arXiv2025

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

A sparse canonical van der Waerden theorem

The canonical van der Waerden theorem asserts that, for sufficiently large $n$, every colouring of $[n]$ contains either a monochromatic or a rainbow arithmetic progression of length $k$ ($k$-AP, for short). In this paper, we determine the threshold at which the binomial random subset $[n]_p$ almost surely inherits this canonical Ramsey type property. As an application, we show the existence of sets $A\subseteq [n]$ such that the $k$-APs in $A$ define a $k$-uniform hypergraph of arbitrarily high girth and yet any colouring of $A$ induces a monochromatic or rainbow $k$-AP.

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-Szemer\'edi (6,3)-Theorem.

math.CO↗

arXiv2025

A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs

A meta-conjecture of Coulson, Keevash, Perarnau and Yepremyan states that above the extremal threshold for a given spanning structure in a (hyper-)graph, one can find a rainbow version of that spanning structure in any suitably bounded colouring of the host (hyper-)graph. We solve one of the most pertinent outstanding cases of this conjecture, by showing that for any $1\leq j\leq k-1$, if $G$ is a $k$-uniform hypergraph above the $j$-degree threshold for a loose Hamilton cycle, then any globally bounded colouring of $G$ contains a rainbow loose Hamilton cycle.

math.CO↗

arXiv2024

A note on finding large transversals efficiently

In an $n \times n$ array filled with symbols, a transversal is a collection of entries with distinct rows, columns and symbols. In this note we show that if no symbol appears more than $\beta n$ times, the array contains a transversal of size $(1-\beta/4-o(1))n$. In particular, if the array is filled with $n$ symbols, each appearing $n$ times (an equi-$n$ square), we get transversals of size $(3/4-o(1))n$. Moreover, our proof gives a deterministic algorithm with polynomial running time, that finds these transversals.

math.CO↗

arXiv2024

A canonical Ramsey theorem for even cycles in random graphs

The celebrated canonical Ramsey theorem of Erd\H{o}s and Rado implies that for $2\leq k\in \mathbb{N}$, any colouring of the edges of $K_n$ with $n$ sufficiently large gives a copy of $C_{2k}$ which has one of three canonical colour patterns: monochromatic, rainbow or lexicographic. In this paper we show that if $p=\omega(n^{-1+1/(2k-1)}\log n)$, then ${\mathbf{G}}(n,p)$ will asymptotically almost surely also have the property that any colouring of its edges induces canonical copies of $C_{2k}$. This determines the threshold for the canonical Ramsey property with respect to even cycles, up to a $\log$ factor.

math.CO↗

arXiv2023

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 Bollob\'as 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↗

arXiv2023

Universality for transversal Hamilton cycles

Let $\mathbf{G}=\{G_1, \ldots, G_m\}$ be a graph collection on a common vertex set $V$ of size $n$ such that $\delta(G_i) \geq (1+o(1))n/2$ for every $i \in [m]$. We show that $\mathbf{G}$ contains every Hamilton cycle pattern. That is, for every map $\chi: [n] \to [m]$ there is a Hamilton cycle whose $i$-th edge lies in $G_{\chi(i)}$.

math.CO↗

arXiv2023

Slow graph bootstrap percolation I: Cycles

Given a fixed graph $H$ and an $n$-vertex graph $G$, the $H$\emph{-bootstrap percolation process} on $G$ is defined to be the sequence of graphs $G_i$, $i\geq 0$ which starts with $G_0 := G$ and in which $G_{i+1}$ is obtained from $G_i$ by adding every edge that completes a copy of $H$. We are interested in $M_H(n)$ which is the maximum number of steps, over all $n$-vertex graphs $G$, that this process takes to stabilise. We determine this maximum running time precisely when $H$ is a cycle, giving the first infinite family of graphs $H$ for which an exact solution is known. We find that $M_{C_k}(n)$ is of order $\log_{k-1}(n)$ for all $3\leq k\in \mathbb{N}$. Interestingly though, the function exhibits different behaviour depending on the parity of $k$ and the exact location of the values of $n$ for which $M_H(n)$ increases is determined by the Frobenius number of a certain numerical semigroup depending on $k$.

math.CO↗

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

Two-round Ramsey games on random graphs

Motivated by the investigation of sharpness of thresholds for Ramsey properties in random graphs, Friedgut, Kohayakawa, R\"odl, Ruci\'nski and Tetali introduced two variants of a single-player game whose goal is to colour the edges of a~random graph, in an online fashion, so as not to create a monochromatic triangle. In the two-round variant of the game, the player is first asked to find a triangle-free colouring of the edges of a random graph $G_1$ and then extend this colouring to a triangle-free colouring of the union of $G_1$ and another (independent) random graph $G_2$, which is disclosed to the player only after they have coloured $G_1$. Friedgut et al.\ analysed this variant of the online Ramsey game in two instances: when $G_1$ has $\Theta(n^{4/3})$ edges and when the number of edges of $G_1$ is just below the threshold above which a random graph typically no longer admits a triangle-free colouring, which is located at $\Theta(n^{3/2})$. The two-round Ramsey game has been recently revisited by Conlon, Das, Lee and M\'esz\'aros, who generalised the result of Friedgut at al.\ from triangles to all strictly $2$-balanced graphs. We extend the work of Friedgut et al.\ in an orthogonal direction and analyse the triangle case of the two-round Ramsey game at all intermediate densities. More precisely, for every $n^{-4/3} \ll p \ll n^{-1/2}$, with the exception of $p = \Theta(n^{-3/5})$, we determine the threshold density $q$ at which it becomes impossible to extend any triangle-free colouring of a typical $G_1 \sim G_{n,p}$ to a triangle-free colouring of the union of $G_1$ and $G_2 \sim G_{n,q}$. An interesting aspect of our result is that this threshold density $q$ `jumps' by a polynomial quantity as $p$ crosses a `critical' window around $n^{-3/5}$.

math.CO↗

arXiv2023

A canonical Ramsey theorem with list constraints in random (hyper-)graphs

The celebrated canonical Ramsey theorem of Erd\H{o}s and Rado implies that for a given $k$-uniform hypergraph (or $k$-graph) $H$, if $n$ is sufficiently large then any colouring of the edges of the complete $k$-graph $K^{(k)}_n$ gives rise to copies of $H$ that exhibit certain colour patterns. We are interested in sparse random versions of this result and the threshold at which the random $k$-graph ${\mathbf{G}}^{(k)}(n,p)$ inherits the canonical Ramsey properties of $K^{(k)}_n$. Our main result here pins down this threshold when we focus on colourings that are constrained by some prefixed lists. This result is applied in an accompanying work of the authors on the threshold for the canonical Ramsey property (with no list constraints) in the case that $H$ is a (2-uniform) even cycle.

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

A robust Corr\'adi--Hajnal Theorem

For a graph $G$ and $p\in[0,1]$, we denote by $G_p$ the random sparsification of $G$ obtained by keeping each edge of $G$ independently, with probability $p$. We show that there exists a $C>0$ such that if $p\geq C(\log n)^{1/3}n^{-2/3}$ and $G$ is an $n$-vertex graph with $n\in 3\mathbb{N}$ and $\delta(G)\geq \tfrac{2n}{3}$, then with high probability $G_p$ contains a triangle factor. Both the minimum degree condition and the probability condition, up to the choice of $C$, are tight. Our result can be viewed as a common strengthening of the seminal theorems of Corr\'adi and Hajnal, which deals with the extremal minimum degree condition for containing triangle factors (corresponding to $p=1$ in our result), and Johansson, Kahn and Vu, which deals with the threshold for the appearance of a triangle factor in $G(n,p)$ (corresponding to $G=K_n$ in our result). It also implies a lower bound on the number of triangle factors in graphs with minimum degree at least $\tfrac{2n}{3}$ which gets close to the truth.

math.CO↗