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David Ellis

Publications and source records attributed to David Ellis.

At least 37 records · Page 2Linked to original sources
arXiv2022

Personality Traits in Game Development

Existing work on personality traits in software development excludes game developers as a discrete group. Whilst games are software, game development has unique considerations, so game developers may exhibit different personality traits from other software professionals. We assessed responses from 123 game developers on an International Personality Item Pool Five Factor Model scale and demographic questionnaire using factor analysis. Programmers reported lower Extraversion than designers, artists and production team members; lower Openness than designers and production, and reported higher Neuroticism than production -- potentially linked to burnout and crunch time. Compared to published norms of software developers, game developers reported lower Openness, Conscientiousness, Extraversion and Agreeableness, but higher Neuroticism. These personality differences have many practical implications: differences in Extraversion among roles may precipitate communication breakdowns; differences in Openness may induce conflict between programmers and designers. Understanding the relationship between personality traits and roles can help recruiters steer new employees into appropriate roles, and help managers apply appropriate stress management techniques. To realise these benefits, individuals must be distinguished from roles: just because an individual occupies a role does not mean they possess personality traits associated with that role.

cs.SE↗
arXiv2021

Pooled testing and its applications in the COVID-19 pandemic

When testing for a disease such as COVID-19, the standard method is individual testing: we take a sample from each individual and test these samples separately. An alternative is pooled testing (or "group testing"), where samples are mixed together in different pools, and those pooled samples are tested. When the prevalence of the disease is low and the accuracy of the test is fairly high, pooled testing strategies can be more efficient than individual testing. In this chapter, we discuss the mathematics of pooled testing and its uses during pandemics, in particular the COVID-19 pandemic. We analyse some one- and two-stage pooling strategies under perfect and imperfect tests, and consider the practical issues in the application of such protocols.

stat.AP↗
arXiv2021

Union-closed families with small average overlap densities

In this very short paper, we point out that the average overlap density of a union-closed family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$ may be as small as $Θ((\log \log |\mathcal{F}|)/(\log |\mathcal{F}|))$, for infinitely many positive integers $n$.

math.CO↗
arXiv2021

Intersection Problems in Extremal Combinatorics: Theorems, Techniques and Questions Old and New

The study of intersection problems in Extremal Combinatorics dates back perhaps to 1938, when Paul Erdős, Chao Ko and Richard Rado proved the (first) `Erdős-Ko-Rado theorem' on the maximum possible size of an intersecting family of $k$-element subsets of a finite set. Since then, a plethora of results of a similar flavour have been proved, for a range of different mathematical structures, using a wide variety of different methods. Structures studied in this context have included families of vector subspaces, families of graphs, subsets of finite groups with given group actions, and of course uniform hypergraphs with stronger or weaker intersection conditions imposed. The methods used have included purely combinatorial ones such as shifting/compressions, algebraic methods (including linear-algebraic, Fourier analytic and representation-theoretic), and more recently, analytic, probabilistic and regularity-type methods. As well as being natural problems in their own right, intersection problems have connections with many other parts of Combinatorics and with Theoretical Computer Science (and indeed with many other parts of Mathematics), both through the results themselves, and the methods used. In this survey paper, we discuss both old and new results (and both old and new methods), in the field of intersection problems. Many interesting open problems remain; we will discuss several. For expositional and pedagogical purposes, we also take this opportunity to give slightly streamlined versions of proofs (due to others) of several classical results in the area. This survey is intended to be useful to PhD students, as well as to more established researchers. It is a personal perspective on the field, and is not intended to be exhaustive; we apologise for any omissions. It is an expanded version of a paper that will appear in the Proceedings of the 29th British Combinatorial Conference.

math.CO↗
arXiv2021

Axion Miniclusters Made Easy

We use a modified version of the Peak Patch excursion set formalism to compute the mass and size distribution of QCD axion miniclusters from a fully non-Gaussian initial density field obtained from numerical simulations of axion string decay. We find strong agreement with N-Body simulations at a significantly lower computational cost. We employ a spherical collapse model and provide fitting functions for the modified barrier in the radiation era. The halo mass function at $z=629$ has a power-law distribution $M^{-0.6}$ for masses within the range $10^{-15}\lesssim M\lesssim 10^{-10}M_{\odot}$, with all masses scaling as $(m_a/50μ\mathrm{eV})^{-0.5}$. We construct merger trees to estimate the collapse redshift and concentration mass relation, $C(M)$, which is well described using analytical results from the initial power spectrum and linear growth. Using the calibrated analytic results to extrapolate to $z=0$, our method predicts a mean concentration $C\sim \mathcal{O}(\text{few})\times10^4$. The low computational cost of our method makes future investigation of the statistics of rare, dense miniclusters easy to achieve.

astro-ph.CO↗
arXiv2020

On the structure of random graphs with constant $r$-balls

We continue the study of the properties of graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to the ball of radius $r$ in some fixed vertex-transitive graph $F$, for various choices of $F$ and $r$. This is a natural extension of the study of regular graphs. More precisely, if $F$ is a vertex-transitive graph and $r \in \mathbb{N}$, we say a graph $G$ is {\em $r$-locally $F$} if the ball of radius $r$ around each vertex of $G$ induces a graph isomorphic to the graph induced by the ball of radius $r$ around any vertex of $F$. We consider the following random graph model: for each $n \in \mathbb{N}$, we let $G_n = G_n(F,r)$ be a graph chosen uniformly at random from the set of all unlabelled, $n$-vertex graphs that are $r$-locally $F$. We investigate the properties possessed by the random graph $G_n$ with high probability, for various natural choices of $F$ and $r$. We prove that if $F$ is a Cayley graph of a torsion-free group of polynomial growth, and $r$ is sufficiently large depending on $F$, then the random graph $G_n = G_n(F,r)$ has largest component of order at most $n^{5/6}$ with high probability, and has at least $\exp(n^δ)$ automorphisms with high probability, where $δ>0$ depends upon $F$ alone. Both properties are in stark contrast to random $d$-regular graphs, which correspond to the case where $F$ is the infinite $d$-regular tree. We also show that, under the same hypotheses, the number of unlabelled, $n$-vertex graphs that are $r$-locally $F$ grows like a stretched exponential in $n$, again in contrast with $d$-regular graphs. In the case where $F$ is the standard Cayley graph of $\mathbb{Z}^d$, we obtain a much more precise enumeration result, and more precise results on the properties of the random graph $G_n(F,r)$. Our proofs use a mixture of results and techniques from geometry, group theory and combinatorics.

math.CO↗
arXiv2019

Approximation by juntas in the symmetric group, and forbidden intersection problems

A family of permutations $\mathcal{F} \subset S_{n}$ is said to be $t$-intersecting if any two permutations in $\mathcal{F}$ agree on at least $t$ points. It is said to be $(t-1)$-intersection-free if no two permutations in $\mathcal{F}$ agree on exactly $t-1$ points. If $S,T \subset \{1,2,\ldots,n\}$ with $|S|=|T|$, and $π: S \to T$ is a bijection, the $π$-star in $S_n$ is the family of all permutations in $S_n$ that agree with $π$ on all of $S$. An $s$-star is a $π$-star such that $π$ is a bijection between sets of size $s$. Friedgut and Pilpel, and independently the first author, showed that if $\mathcal{F} \subset S_n$ is $t$-intersecting, and $n$ is sufficiently large depending on $t$, then $|\mathcal{F}| \leq (n-t)!$; this proved a conjecture of Deza and Frankl from 1977. Equality holds only if $\mathcal{F}$ is a $t$-star. In this paper, we give a more `robust' proof of a strengthening of the Deza-Frankl conjecture, namely that if $n$ is sufficiently large depending on $t$, and $\mathcal{F} \subset S_n$ is $(t-1)$-intersection-free, then $|\mathcal{F} \leq (n-t)!$, with equality only if $\mathcal{F}$ is a $t$-star. The main ingredient of our proof is a `junta approximation' result, namely, that any $(t-1)$-intersection-free family of permutations is essentially contained in a $t$-intersecting {\em junta} (a `junta' being a union of a bounded number of $O(1)$-stars). The proof of our junta approximation result relies, in turn, on a weak regularity lemma for families of permutations, a combinatorial argument that `bootstraps' a weak notion of pseudorandomness into a stronger one, and finally a spectral argument for pairs of highly-pseudorandom fractional families. Our proof employs four different notions of pseudorandomness, three being combinatorial in nature, and one being algebraic.

math.CO↗
arXiv2019

Setwise intersecting families of permutations

A family of permutations $A \subset S_n$ is said to be \emph{$t$-set-intersecting} if for any two permutations $σ, π\in A$, there exists a $t$-set $x$ whose image is the same under both permutations, i.e. $σ(x)=π(x)$. We prove that if $n$ is sufficiently large depending on $t$, the largest $t$-set-intersecting families of permutations in $S_n$ are cosets of stabilizers of $t$-sets. The $t=2$ case of this was conjectured by János Körner. It can be seen as a variant of the Deza-Frankl conjecture, proved in [4]. Our proof uses similar techniques to those of [4], namely, eigenvalue methods, together with the representation theory of the symmetric group, but the combinatorial part of the proof is harder.

math.CO↗
arXiv2019

Smallest cyclically covering subspaces of $\mathbb{F}_q^n$, and lower bounds in Isbell's conjecture

For a prime power $q$ and a positive integer $n$, we say a subspace $U$ of ${\mathbb{F}_q^n}$ is {\em cyclically covering} if the union of the cyclic shifts of $U$ is equal to $\mathbb{F}_q^n$. We investigate the problem of determining the minimum possible dimension of a cyclically covering subspace of $\mathbb{F}_q^n$. (This is a natural generalisation of a problem posed in 1991 by the first author.) We prove several upper and lower bounds, and for each fixed $q$, we answer the question completely for infinitely many values of $n$ (which take the form of certain geometric series). Our results imply lower bounds for a well-known conjecture of Isbell, and a generalisation theoreof, supplementing lower bounds due to Spiga. We also consider the analogous problem for general representations of groups. We use arguments from combinatorics, representation theory and finite field theory.

math.CO↗
arXiv2018

On the union of intersecting families

A family of sets is said to be \emph{intersecting} if any two sets in the family have nonempty intersection. In 1973, Erdős raised the problem of determining the maximum possible size of a union of $r$ different intersecting families of $k$-element subsets of an $n$-element set, for each triple of integers $(n,k,r)$. We make progress on this problem, proving that for any fixed integer $r \geq 2$ and for any $k \leq (\tfrac{1}{2}-o(1))n$, if $X$ is an $n$-element set, and $\mathcal{F} = \mathcal{F}_1 \cup \mathcal{F}_2 \cup \ldots \cup \mathcal{F}_r$, where each $\mathcal{F}_i$ is an intersecting family of $k$-element subsets of $X$, then $|\mathcal{F}| \leq {n \choose k} - {n-r \choose k}$, with equality only if $\mathcal{F} = \{S \subset X:\ |S|=k,\ S \cap R \neq \emptyset\}$ for some $R \subset X$ with $|R|=r$. This is best possible up to the size of the $o(1)$ term, and improves a 1987 result of Frankl and Füredi, who obtained the same conclusion under the stronger hypothesis $k < (3-\sqrt{5})n/2$, in the case $r=2$. Our proof utilises an isoperimetric, influence-based method recently developed by Keller and the authors.

math.CO↗
arXiv2018

Stability versions of Erdős-Ko-Rado type theorems, via isoperimetry

Erdős-Ko-Rado (EKR) type theorems yield upper bounds on the sizes of families of sets, subject to various intersection requirements on the sets in the family. Stability versions of such theorems assert that if the size of a family is close to the maximum possible size, then the family itself must be close (in some appropriate sense) to a maximum-sized family. In this paper, we present an approach to obtaining stability versions of EKR-type theorems, via isoperimetric inequalities for subsets of the hypercube. Our approach is rather general, and allows the leveraging of a wide variety of exact EKR-type results into strong stability versions of these results, without going into the proofs of the original results. We use this approach to obtain tight stability versions of the EKR theorem itself and of the Ahlswede-Khachatrian theorem on $t$-intersecting families of $k$-element subsets of $\{1,2,\ldots.n\}$ (for $k < \frac{n}{t+1}$), and to show that, somewhat surprisingly, all these results hold when the intersection requirement is replaced by a much weaker requirement. Other examples include stability versions of Frankl's recent result on the Erdős matching conjecture, the Ellis-Filmus-Friedgut proof of the Simonovits-Sós conjecture, and various EKR-type results on $r$-wise (cross)-$t$-intersecting families.

math.CO↗
arXiv2018

On the structure of subsets of the discrete cube with small edge boundary

The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers $m$ and $n$, the minimum size $g_n(m)$ of the edge boundary of an $m$-element subset of $\{0,1\}^{n}$; the extremal families (up to automorphisms of the discrete cube) are initial segments of the lexicographic ordering on $\{0,1\}^n$. We show that for any $m$-element subset $\mathcal{F} \subset \{0,1\}^n$ and any integer $l$, if the edge boundary of $\mathcal{F}$ has size at most $g_n(m)+l$, then there exists an extremal family $\mathcal{G} \subset \{0,1\}^n$ such that $|\mathcal{F} Δ\mathcal{G}| \leq Cl$, where $C$ is an absolute constant. This is best-possible, up to the value of $C$. Our result can be seen as a `stability' version of the edge isoperimetric inequality in the discrete cube, and as a discrete analogue of the seminal stability result of Fusco, Maggi and Pratelli concerning the isoperimetric inequality in Euclidean space.

math.CO↗
arXiv2018

Stability for the Complete Intersection Theorem, and the Forbidden Intersection Problem of Erdős and Sós

A family $F$ of sets is said to be $t$-intersecting if $|A \cap B| \geq t$ for any $A,B \in F$. The seminal Complete Intersection Theorem of Ahlswede and Khachatrian (1997) gives the maximal size $f(n,k,t)$ of a $t$-intersecting family of $k$-element subsets of $[n]=\{1,2,\ldots,n\}$, together with a characterisation of the extremal families. The forbidden intersection problem, posed by Erdős and Sós in 1971, asks for a determination of the maximal size $g(n,k,t)$ of a family $F$ of $k$-element subsets of $[n]$ such that $|A \cap B| \neq t-1$ for any $A,B \in F$. In this paper, we show that for any fixed $t \in \mathbb{N}$, if $o(n) \leq k \leq n/2-o(n)$, then $g(n,k,t)=f(n,k,t)$. In combination with prior results, this solves the above problem of Erdős and Sós for any constant $t$, except for in the ranges $n/2-o(n) < k < n/2+t/2$ and $k < 2t$. One key ingredient of the proof is the following sharp `stability' result for the Complete Intersection Theorem: if $k/n$ is bounded away from $0$ and $1/2$, and $F$ is a $t$-intersecting family of $k$-element subsets of $[n]$ such that $|F| \geq f(n,k,t) - O(\binom{n-d}{k})$, then there exists a family $G$ such that $G$ is extremal for the Complete Intersection Theorem, and $|F \setminus G| = O(\binom{n-d}{k-d})$. We believe this result to be of interest in its own right; indeed, it proves a conjecture of Friedgut from 2008. We prove it by combining classical `shifting' arguments with a `bootstrapping' method based upon an isoperimetric inequality. Another key ingredient is a `weak regularity lemma' for families of $k$-element subsets of $[n]$, where $k/n$ is bounded away from 0 and 1. This states that any such family $F$ is approximately contained within a `junta', such that the restriction of $F$ to each subcube determined by the junta is `pseudorandom' in a certain sense.

math.CO↗
arXiv2018

On a biased edge isoperimetric inequality for the discrete cube

The `full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary $\partial A$ of a set $A \subset \{0,1\}^n$, as a function of $|A|$. A weaker (but more widely-used) lower bound is $|\partial A| \geq |A| \log_2(2^n/|A|)$, where equality holds iff $A$ is a subcube. In 2011, the first author obtained a sharp `stability' version of the latter result, proving that if $|\partial A| \leq |A| (\log(2^n/|A|)+ε)$, then there exists a subcube $C$ such that $|A ΔC|/|A| = O(ε/\log(1/ε))$. The `weak' version of the edge isoperimetric inequality has the following well-known generalization for the `$p$-biased' measure $μ_p$ on the discrete cube: if $p \leq 1/2$, or if $0 < p < 1$ and $A$ is monotone increasing, then $pμ_p(\partial A) \geq μ_p(A) \log_p(μ_p(A))$. In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if $pμ_p(\partial A) \leq μ_p(A) (\log_p(μ_p(A))+ε)$, then there exists a subcube $C$ such that $μ_p(A ΔC)/μ_p(A) = O(ε' /\log(1/ε'))$, where $ε' =ε\ln (1/p)$. This result is a central component in recent work of the authors proving sharp stability versions of a number of Erdős-Ko-Rado type theorems in extremal combinatorics, including the seminal `complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the `full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the `full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem.

math.CO↗
arXiv2017

An isoperimetric inequality for antipodal subsets of the discrete cube

A family of subsets of $\{1,2,\ldots,n\}$ is said to be {\em antipodal} if it is closed under taking complements. We prove a best-possible isoperimetric inequality for antipodal families of subsets of $\{1,2,\ldots,n\}$. Our inequality implies that for any $k \in \mathbb{N}$, among all such families of size $2^k$, a family consisting of the union of a $(k-1)$-dimensional subcube and its antipode has the smallest possible edge boundary.

math.CO↗
arXiv2017

On regular hypergraphs of high girth

We give lower bounds on the maximum possible girth of an $r$-uniform, $d$-regular hypergraph with at most $n$ vertices, using the definition of a hypergraph cycle due to Berge. These differ from the trivial upper bound by an absolute constant factor (viz., by a factor of between $3/2+o(1)$ and $2 +o(1)$). We also define a random $r$-uniform `Cayley' hypergraph on $S_n$ which has girth $Ω(n^{1/3})$ with high probability, in contrast to random regular $r$-uniform hypergraphs, which have constant girth with positive probability.

math.CO↗
arXiv2017

Intersecting Families of Permutations

A set of permutations $I \subset S_n$ is said to be {\em k-intersecting} if any two permutations in $I$ agree on at least $k$ points. We show that for any $k \in \mathbb{N}$, if $n$ is sufficiently large depending on $k$, then the largest $k$-intersecting subsets of $S_n$ are cosets of stabilizers of $k$ points, proving a conjecture of Deza and Frankl. We also prove a similar result concerning $k$-cross-intersecting subsets. Our proofs are based on eigenvalue techniques and the representation theory of the symmetric group.

math.CO↗