arXiv subjects
Publications and source records attributed to David Ellis.
In this note we show that if the sphere $\mathbb{S}^n$ is covered by $k$ open sets with $n \geq 2k-2$, then one of these sets contains a path with antipodal endpoints. This is best possible in the sense that the statement fails for $n < 2k-2$. The result can be seen as a spherical analogue of a well-known conjecture of Norine on edge-colourings of the discrete hypercube.
Stress remains a significant social problem for individuals in modern societies. This paper presents a machine learning approach for the automatic detection of stress of people in a social situation by combining two sensor systems that capture physiological and social responses. We compare the performance using different classifiers including support vector machine, AdaBoost, and k-nearest neighbor. Our experimental results show that by combining the measurements from both sensor systems, we could accurately discriminate between stressful and neutral situations during a controlled Trier social stress test (TSST). Moreover, this paper assesses the discriminative ability of each sensor modality individually and considers their suitability for real-time stress detection. Finally, we present an study of the most discriminative features for stress detection.
The proportion of $d$-element subsets of $\mathbb{F}_2^d$ that are bases is asymptotic to $\prod_{j=1}^{\infty}(1-2^{-j}) \approx 0.29$ as $d \to \infty$. It is natural to ask whether there exists a (large) subset $\mathcal{F}$ of $\mathbb{F}_2^d$ such that the proportion of $d$-element subsets of $\mathcal{F}$ that are bases is (asymptotically) greater than this number. As well as being a natural question in its own right, this would imply better lower bounds on the Tur\'an densities of certain hypercubes and `daisy' hypergraphs. We give a negative answer to the above question. More generally, we obtain an asymptotically sharp upper bound on the proportion of linearly independent $r$-element subsets of a (large) family of vectors in $\mathbb{F}_2^d$, for $r \leq d$. This bound follows from an exact result concerning the probability of obtaining a linearly independent sequence when we randomly sample $r$ elements with replacement from our family of vectors: we show that this probability, for any family of vectors, is at most what it is when the family is the whole space $\mathbb{F}_2^d \setminus \{0\}$. Our results also go through when $\mathbb{F}_2$ is replaced by $\mathbb{F}_q$ for any prime power $q$.
How small can a set of vertices in the $n$-dimensional hypercube $Q_n$ be if it meets every copy of $Q_d$? The asymptotic density of such a set (for $d$ fixed and $n$ large) is denoted by $\gamma_d$. It is easy to see that $\gamma_d \leq 1/(d+1)$, and it is known that $\gamma_d=1/(d+1)$ for $d \leq 2$, but it was recently shown that $\gamma_d < 1/(d+1)$ for $d \geq 8$. In this paper we show that the latter phenomenon also holds for $d=7$ and $d=6$.
We show that if $\mathcal{A} \subset {[n] \choose n/2}$ with measure bounded away from zero and from one, then the $\Omega(\sqrt{n})$-iterated upper shadows of $\mathcal{A}$ and $\mathcal{A}^c$ intersect in a set of positive measure. This confirms (in a strong form) a conjecture of Friedgut. It can be seen as a stability result for the Kruskal--Katona theorem.
Let $G$ be a nontrivial permutation group of degree $n$. If $G$ is transitive, then a theorem of Jordan states that $G$ has a derangement. Equivalently, a finite group is never the union of conjugates of a proper subgroup. If $G$ is intransitive, then $G$ may fail to have a derangement, and this can happen even if $G$ has only two orbits, both of which have size $(1/2+o(1))n$. However, we conjecture that if $G$ has two orbits of size exactly $n/2$ then $G$ does have a derangement, and we prove this conjecture when $G$ acts primitively on at least one of the orbits. Equivalently, we conjecture that a finite group is never the union of conjugates of two proper subgroups of the same order, and we prove this conjecture when at least one of the subgroups is maximal. (Feldman also implicitly raised this conjecture on StackExchange.) We also prove the conjecture for soluble groups, almost simple groups and groups of order at most 50000, and we reduce the conjecture to perfect groups. Along the way, we prove a linear variant on Isbell's conjecture regarding derangements of prime-power order, and we highlight connections with intersecting families of permutations and roots of polynomials modulo primes.
An $r$-daisy is an $r$-uniform hypergraph consisting of the six $r$-sets formed by taking the union of an $(r-2)$-set with each of the 2-sets of a disjoint 4-set. Bollob\'as, Leader and Malvenuto, and also Bukh, conjectured that the Tur\'an density of the $r$-daisy tends to zero as $r \to \infty$. In this paper we disprove this conjecture. Adapting our construction, we are also able to disprove a folklore conjecture about Tur\'an densities of hypercubes. For fixed $d$ and large $n$, we show that the smallest set of vertices of the $n$-dimensional hypercube $Q_n$ that meets every copy of $Q_d$ has asymptotic density strictly below $1/(d+1)$, for all $d \geq 8$. In fact, we show that this asymptotic density is at most $c^d$, for some constant $c<1$. As a consequence, we obtain similar bounds for the edge-Tur\'an densities of hypercubes. We also answer some related questions of Johnson and Talbot, and disprove a conjecture made by Bukh and by Griggs and Lu on poset densities.
If $G$ is a group, we say a subset $S$ of $G$ is product-free if the equation $xy=z$ has no solutions with $x,y,z \in S$. For $D \in \mathbb{N}$, a group $G$ is said to be $D$-quasirandom if the minimal dimension of a nontrivial complex irreducible representation of $G$ is at least $D$. Gowers showed that in a $D$-quasirandom finite group $G$, the maximal size of a product-free set is at most $|G|/D^{1/3}$. This disproved a longstanding conjecture of Babai and S\'os from 1985. For the special unitary group, $G=SU(n)$, Gowers observed that his argument yields an upper bound of $n^{-1/3}$ on the measure of a measurable product-free subset. In this paper, we improve Gowers' upper bound to $\exp(-cn^{1/3})$, where $c>0$ is an absolute constant. In fact, we establish something stronger, namely, product-mixing for measurable subsets of $SU(n)$ with measure at least $\exp(-cn^{1/3})$; for this product-mixing result, the $n^{1/3}$ in the exponent is sharp. Our approach involves introducing novel hypercontractive inequalities, which imply that the non-Abelian Fourier spectrum of the indicator function of a small set concentrates on high-dimensional irreducible representations. Our hypercontractive inequalities are obtained via methods from representation theory, harmonic analysis, random matrix theory and differential geometry. We generalize our hypercontractive inequalities from $SU(n)$ to an arbitrary $D$-quasirandom compact connected Lie group for $D$ at least an absolute constant, thereby extending our results on product-free sets to such groups. We also demonstrate various other applications of our inequalities to geometry (viz., non-Abelian Brunn-Minkowski type inequalities), mixing times, and the theory of growth in compact Lie groups.
For integers $n \geq k \geq 1$, the {\em Kneser graph} $K(n, k)$ is the graph with vertex-set consisting of all the $k$-element subsets of $\{1,2,\ldots,n\}$, where two $k$-element sets are adjacent in $K(n,k)$ if they are disjoint. We show that if $(n,k,s) \in \mathbb{N}^3$ with $n > 10000 k s^5$ and $\mathcal{F}$ is set of vertices of $K(n,k)$ of size larger than $\{A \subset \{1,2,\ldots,n\}:\ |A|=k,\ A \cap \{1,2,\ldots,s\} \neq \varnothing\}$, then the subgraph of $K(n,k)$ induced by $\mathcal{F}$ has maximum degree at least \[ \left(1 - O\left(\sqrt{s^3 k/n}\right)\right)\frac{s}{s+1} \cdot {n-k \choose k} \cdot \frac{|\mathcal{F}|}{\binom{n}{k}}.\] This is sharp up to the behaviour of the error term $O(\sqrt{s^3 k/n})$. In particular, if the triple of integers $(n, k, s)$ satisfies the condition above, then the minimum maximum degree does not increase `continuously' with $|\mathcal{F}|$. Instead, it has $s$ jumps, one at each time when $|\mathcal{F}|$ becomes just larger than the union of $i$ stars, for $i = 1, 2, \ldots, s$. An appealing special case of the above result is that if $\mathcal{F}$ is a family of $k$-element subsets of $\{1,2,\ldots,n\}$ with $|\mathcal{F}| = {n-1 \choose k-1}+1$, then there exists $A \in \mathcal{F}$ such that $\mathcal{F}$ is disjoint from at least $$\left(1/2-O\left(\sqrt{k/n}\right)\right){n-k-1 \choose k-1}$$ of the other sets in $\mathcal{F}$; this is asymptotically sharp if $k=o(n)$. Frankl and Kupavskii, using different methods, have recently proven closely related results under the hypothesis that $n$ is at least quadratic in $k$.
One of the great pleasures of working with Imre Leader is to experience his infectious delight on encountering a compelling combinatorial problem. This collection of open problems in combinatorics has been put together by a subset of his former PhD students and students-of-students for the occasion of his 60th birthday. All of the contributors have been influenced (directly or indirectly) by Imre: his personality, enthusiasm and his approach to mathematics. The problems included cover many of the areas of combinatorial mathematics that Imre is most associated with: including extremal problems on graphs, set systems and permutations, and Ramsey theory. This is a personal selection of problems which we find intriguing and deserving of being better known. It is not intended to be systematic, or to consist of the most significant or difficult questions in any area. Rather, our main aim is to celebrate Imre and his mathematics and to hope that these problems will make him smile. We also hope this collection will be a useful resource for researchers in combinatorics and will stimulate some enjoyable collaborations and beautiful mathematics.
In this very short note, we give a counterexample to a recent conjecture of Gilmer which would have implied the union-closed conjecture.
We prove an analogue of Bonami's (hypercontractive) lemma for complex-valued functions on $\mathcal{L}(V,W)$, where $V$ and $W$ are vector spaces over a finite field. This inequality is useful for functions on $\mathcal{L}(V,W)$ whose `generalised influences' are small, in an appropriate sense. It leads to a significant shortening of the proof of a recent seminal result by Khot, Minzer and Safra that pseudorandom sets in Grassmann graphs have near-perfect expansion, which (in combination with the work of Dinur, Khot, Kindler, Minzer and Safra) implies the 2-2 Games conjecture (the variant, that is, with imperfect completeness).
We study an analogue of the Erd\H{o}s-S\'os forbidden intersection problem, for families of linear maps. If $V$ and $W$ are vector spaces over the same field, we say a family $\mathcal{F}$ of linear maps from $V$ to $W$ is \emph{$(t-1)$-intersection-free} if for any two linear maps $\sigma_1,\sigma_2 \in \mathcal{F}$, $\dim(\{v \in V:\ \sigma_1(v)=\sigma_2(v)\}) \neq t-1$. We prove that if $n$ is sufficiently large depending on $t$, $q$ is any prime power, $V$ is an $n$-dimensional vector space over $\mathbb{F}_q$, and $\mathcal{F} \subset \textrm{GL}(V)$ is $(t-1)$-intersection-free, then $|\mathcal{F}| \leq \prod_{i=1}^{n-t}(q^n - q^{i+t-1})$. Equality holds only if there exists a $t$-dimensional subspace of $V$ on which all elements of $\mathcal{F}$ agree, or a $t$-dimensional subspace of $V^*$ on which all elements of $\{\sigma^*:\ \sigma \in \mathcal{F}\}$ agree. Our main tool is a `junta approximation' result for families of linear maps with a forbidden intersection: namely, that if $V$ and $W$ are finite-dimensional vector spaces over the same finite field, then any $(t-1)$-intersection-free family of linear maps from $V$ to $W$ is essentially contained in a $t$-intersecting \emph{junta} (meaning, a family $\mathcal{J}$ of linear maps from $V$ to $W$ such that the membership of $\sigma$ in $\mathcal{J}$ is determined by $\sigma(v_1),\ldots,\sigma(v_M),\sigma^*(a_1),\ldots,\sigma^*(a_N)$, where $v_1,\ldots,v_M \in V$, $a_1,\ldots,a_N \in W^*$ and $M+N$ is bounded). The proof of this in turn relies on a variant of the `junta method' (originally introduced by Dinur and Friedgut, and powefully extended by Keller and the last author), together with spectral techniques and a hypercontractive inequality.
The Peak-Patch algorithm is used to identify the densest minicluster seeds in the initial axion density field simulated from string decay. The fate of these dense seeds is found by tracking the subsequent gravitational collapse in cosmological $N$-body simulations. We find that miniclusters at late times are well described by NFW profiles, although for around 80\% of simulated miniclusters a single power-law density profile of $r^{-2.9}$ is an equally good fit due to the unresolved scale radius. Under the assumption that all miniclusters with an unresolved scale radius are described by a power-law plus axion star density profile, we identify a significant number of miniclusters that might be dense enough to give rise to gravitational microlensing if the axion mass is $0.2 \,\mathrm{meV}\lesssim m_a \lesssim 3\,\mathrm{meV}$. Higher resolution simulations resolving the inner structure and axion star formation are necessary to explore this possibility further.
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.
For integers $r \geq 3$ and $t \geq 2$, an $r$-uniform $t$-daisy $\mathcal{D}^t_r$ is a family of $\binom{2t}{t}$ $r$-element sets of the form $$\{S \cup T \ : T\subset U, \ |T|=t \}$$ for some sets $S,U$ with $|S|=r-t$, $|U|=2t$ and $S \cap U = \emptyset$. It was conjectured by Bollob\'as, Leader and Malvenuto (and independently Bukh) that the Tur\'an densities of $t$-daisies satisfy $\lim\limits_{r \to \infty} \pi(\mathcal{D}_r^t) = 0$ for all $t \geq 2$; this has become a well-known problem, and it is still open for all values of $t$. In this paper, we give lower bounds for the Tur\'an densities of $r$-uniform $t$-daisies. To do so, we introduce (and make some progress on) the following natural problem in additive combinatorics: for integers $m \geq 2t \geq 4$, what is the maximum cardinality $g(m,t)$ of a subset $R$ of $\mathbb{Z}/m\mathbb{Z}$ such that for any $x \in \mathbb{Z}/m\mathbb{Z}$ and any $2t$-element subset $X$ of $\mathbb{Z}/m\mathbb{Z}$, there are $t$ distinct elements of $X$ whose sum is not in the translate $x+R$? This is a slice-analogue of the extremal Hilbert cube problem considered by Gunderson and R\"odl and its generalization studied by Cilleruelo and Tesoro.
Our aim in this note is to show that, for any $\epsilon>0$, there exists a union-closed family $\mathcal F$ with (unique) smallest set $S$ such that no element of $S$ belongs to more than a fraction $\epsilon$ of the sets in $\mathcal F$. More precisely, we give an example of a union-closed family with smallest set of size $k$ such that no element of this set belongs to more than a fraction $(1+o(1))\frac{\log_2 k}{2k}$ of the sets in $\mathcal F$. We also give explicit examples of union-closed families containing `small' sets for which we have been unable to verify the Union-Closed Conjecture.
The study of intersection problems in Extremal Combinatorics dates back perhaps to 1938, when Paul Erd\H{o}s, Chao Ko and Richard Rado proved the (first) `Erd\H{o}s-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.