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A conjecture of Verstra\"ete states that for any fixed $\ell < k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\ell}$-free subgraph with at least $c |E(G)|$ edges. For $\ell = 2$, this conjecture was verified by K\"uhn and Osthus. We show that $C_6$ and $C_{2k}$ satisfy the conjecture for all odd $k$, but observe that a recent construction of a dense $C_{10}$-free subgraph of the hypercube yields a counterexample to the conjecture for $C_8$ and $C_{10}$., Comment: Our main result follows from the work of K\"uhn and Osthus [J. Graph Theory 48 (2005), 147--156]. In particular, our Theorem 1.4 can be proved using their Lemma 10, with g=6

Given a graph $H$ and a natural number $n$, the extremal number $\mathrm{ex}(n, H)$ is the largest number of edges in an $n$-vertex graph containing no copy of $H$. In this paper, we obtain a general upper bound for the extremal number of generalised face-incidence graphs, a family which includes the standard face-incidence graphs of regular polytopes. This builds on and generalises work of Janzer and Sudakov, who obtained the same bound for hypercubes and bipartite Kneser graphs, and allows us to confirm a conjecture of Conlon and Lee on the extremal number of $K_{r,r}$-free bipartite graphs for certain incidence graphs. In their work, Janzer and Sudakov showed that such an upper bound on the extremal number holds whenever the graph $H$ satisfies a certain percolation property which captures an appropriate sequence of repeated applications of the Cauchy--Schwarz inequality, a property which they then verify for hypercubes and bipartite Kneser graphs. This percolation property bears close resemblance to a property that arose in earlier work of Conlon and Lee on weakly norming graphs. In this latter work, Conlon and Lee developed a method for controlling repeated applications of the Cauchy--Schwarz inequality based on the properties of reflection groups, which then allowed them to isolate a broad family of weakly norming graphs. Here, we develop this method further, casting it in a purely algebraic form that allows us not only to combine it with the Janzer--Sudakov result and obtain the desired result about the extremal number of incidence graphs, but also to simplify the proofs of both the Conlon--Lee result on weakly norming graphs and a related result of Coregliano., Comment: 19 pages, 2 figures

We prove that a random Cayley graph on a group of order $N$ has clique number $O(\log N \log \log N)$ with high probability. This bound is best possible up to the constant factor for certain groups, including~$\mathbb{F}_2^n$, and improves the longstanding upper bound of $O(\log^2 N)$ due to Alon. Our proof does not make use of the underlying group structure and is purely combinatorial, with the key result being an essentially best possible upper bound for the number of subsets of given order that contain at most a given number of colors in a properly edge-colored complete graph. As a further application of this result, we study a conjecture of Alon stating that every group of order $N$ has a Cayley graph whose clique number and independence number are both $O(\log N)$, proving the conjecture for all abelian groups of order $N$ for almost all $N$. For finite vector spaces of order $N$ with characteristic congruent to $1 \pmod 4$, we prove the existence of a self-complementary Cayley graph on the vector space whose clique number and independence number are both at most $(2+o(1))\log N$. This matches the lower bound for Ramsey numbers coming from random graphs and solves, in a strong form, a problem of Alon and Orlitsky motivated by information theory.

This paper is concerned with lattice field models in dimension at least 2. The action is a uniformly convex function of the gradient of the field. The main result Theorem 1.4 proves that charge-charge correlations in the Coulomb dipole gas are close to Gaussian. These results go beyond previous results of Dimock-Hurd and Conlon-Spencer. The approach in the paper is based on the observation that the sine-Gordon probability measure corresponding to the dipole gas is the invariant measure for a certain stochastic dynamics. The stochastic dynamics here differs from the stochastic dynamics in previous work used to study the problem., Comment: 33 pages

A system of linear equations $L$ is said to be norming if a natural functional $t_L(\cdot)$ giving a weighted count for the set of solutions to the system can be used to define a norm on the space of real-valued functions on $\mathbb{F}_q^n$ for every $n>0$. For example, Gowers uniformity norms arise in this way. In this paper, we initiate the systematic study of norming linear systems by proving a range of necessary and sufficient conditions for a system to be norming. Some highlights include an isomorphism theorem for the functional $t_L(\cdot)$, a proof that any norming system must be variable-transitive and the classification of all norming systems of rank at most two., Comment: 30 pages

A natural open problem in Ramsey theory is to determine those $3$-graphs $H$ for which the off-diagonal Ramsey number $r(H, K_n^{(3)})$ grows polynomially with $n$. We make substantial progress on this question by showing that if $H$ is tightly connected or has at most two tight components, then $r(H, K_n^{(3)})$ grows polynomially if and only if $H$ is not contained in an iterated blowup of an edge., Comment: 12 pages

For $p$ prime, $A \subseteq \mathbb{Z}/p\mathbb{Z}$ and $\lambda \in \mathbb{Z}$, the sum of dilates $A + \lambda \cdot A$ is defined by \[A + \lambda \cdot A = \{a + \lambda a' : a, a' \in A\}.\] The basic problem on such sums of dilates asks for the minimum size of $|A + \lambda \cdot A|$ for given $\lambda$, $A$ of given density $\alpha$, and $p$ tending to infinity. We investigate this problem for $\alpha$ fixed and $\lambda$ tending to infinity, proving near-optimal bounds in this case., Comment: 9 pages. To appear in The American Mathematical Monthly

This note expands on the details of the relationship between 4d kination solutions in string cosmology and Kasner solutions of 10d general relativity. It extends previous analyses of this relationship in IIB string theory to other string theories and also to 11d M-theory, while also providing extensive detail on the relationship between perturbations of the 10d Kasner metric and the presence of radiation and matter backgrounds in the dimensionally reduced 4d kination theory.

Gaussian channel simulation is an essential paradigm in understanding the evolution of bosonic quantum states. It allows us to investigate how such states are influenced by the environment and how they transmit quantum information. This makes it an essential tool for understanding the properties of Gaussian quantum communication. Quantum teleportation provides an avenue to effectively simulate Gaussian channels such as amplifier channels, loss channels and classically additive noise channels. However, implementations of these channels, particularly quantum amplifier channels and channels capable of performing Gaussian noise suppression are limited by experimental imperfections and non-ideal entanglement resources. In this work, we overcome these difficulties using a heralded quantum teleportation scheme that is empowered by a measurement-based noiseless linear amplifier. The noiseless linear amplification enables us to simulate a range of Gaussian channels that were previously inaccessible. In particular, we demonstrate the simulation of non-physical Gaussian channels otherwise inaccessible using conventional means. We report Gaussian noise suppression, effectively converting an imperfect quantum channel into a near-identity channel. The performance of Gaussian noise suppression is quantified by calculating the transmitted entanglement., Comment: 9 pages, 4 figures

One of the fundamental tenets of quantum mechanics is that non-orthogonal states cannot be distinguished perfectly. When distinguishing multiple copies of a mixed quantum state, a collective measurement, which generates entanglement between the different copies of the unknown state, can achieve a lower error probability than non-entangling measurements. The error probability that can be attained using a collective measurement on a finite number of copies of the unknown state is given by the Helstrom bound. In the limit where we can perform a collective measurement on asymptotically many copies of the quantum state, the quantum Chernoff bound gives the attainable error probability. It is natural to ask at what rate does the error tend to this asymptotic limit, and whether the asymptotic limit can be attained for any finite number of copies. In this paper we address these questions. We find analytic expressions for the Helstrom bound for arbitrarily many copies of the unknown state in several simple qubit examples. Using these analytic expressions, we investigate how the attainable error rate changes as we allow collective measurements on finite numbers of copies of the quantum state. We also investigate the necessary conditions to saturate the M-copy Helstrom bound. It is known that a collective measurement on all M-copies of the unknown state is always sufficient to saturate the M-copy Helstrom bound. However, general conditions for when such a measurement is necessary to saturate the Helstrom bound remain unknown. We investigate specific measurement strategies which involve entangling operations on fewer than all M-copies of the unknown state. For many regimes we find that a collective measurement on all M-copies of the unknown state is necessary to saturate the M-copy Helstrom bound., Comment: Comments very welcome

Measurement estimation bounds for local quantum multiparameter estimation, which provide lower bounds on possible measurement uncertainties, have so far been formulated in two ways: by extending the classical Cram\'er--Rao bound (e.g., the quantum Cram\'er--Rao bound and the Nagaoka Cram'er--Rao bound) and by incorporating the parameter estimation framework with the uncertainty principle, as in the Lu--Wang uncertainty relation. In this work, we present a general framework that allows a direct comparison between these different types of estimation limits. Specifically, we compare the attainability of the Nagaoka Cram\'er--Rao bound and the Lu--Wang uncertainty relation, using analytical and numerical techniques. We show that these two limits can provide different information about the physically attainable precision. We present an example where both limits provide the same attainable precision and an example where the Lu--Wang uncertainty relation is not attainable even for pure states. We further demonstrate that the unattainability in the latter case arises because the figure of merit underpinning the Lu--Wang uncertainty relation (the difference between the quantum and classical Fisher information matrices) does not necessarily agree with the conventionally used figure of merit (mean squared error). The results offer insights into the general attainability and applicability of the Lu--Wang uncertainty relation. Furthermore, our proposed framework for comparing bounds of different types may prove useful in other settings., Comment: 13 pages, 7 figures

Addition of photons to coherent states is shown to produce effects that display remarkable similarities with cubic phase shifts acting on the vacuum state, with recorded fidelities in excess of 90 percent. The strength of the cubic interaction is found to vary inversely with the displacement of the coherent state and the strongest interactions were one order of magnitude greater than previous observations. The interaction is non-perturbative., Comment: 5 pages, comments very welcome

Herding behavior has become a familiar phenomenon to investors, with potential dangers of both undervaluing and overvaluing assets, while also threatening market stability. This study contributes to the literature on herding behavior by using a recent dataset, covering the most impactful events of recent years. To our knowledge, this is the first study examining herding behavior across three different types of investment vehicle and also the first study observing herding at a community (subset) level. Specifically, we first explore this phenomenon in each separate type of investment vehicle, namely stocks, US ETFs and cryptocurrencies, using the Cross-Sectional Absolute Deviation model. We find mostly similar herding patterns for stocks and US ETFs. Subsequently, the same experiment is implemented on a combination of all three investment vehicles. For a deeper investigation, we adopt graph-based techniques including the Minimum Spanning Tree and Louvain community detection to partition the combination into smaller subsets to detect herding behavior for each subset. We find that herding behavior exists at all times across all types of investment vehicle at a subset level, although perhaps not at the superset level, and that this herding behavior tends to stem from specific events that solely impact that subset of assets. Lastly, we explore herding by examining the financial contagion effects between these types of investment vehicle. Results show that US ETFs not only have a tendency to propagate similar trading behaviors in stocks and especially cryptocurrencies but also show self-reinforcing herding behavior, acting as drivers of their own trends.

We describe a new mechanism, whose ingredients are realised in string compactifications, for the formation of cosmic (super)string networks. Oscillating string loops grow when their tension $\mu$ decreases with time. If $2H + \dot{\mu}/\mu < 0$, where $H$ is the Hubble parameter, loops grow faster than the scale factor and an initial population of isolated small loops (for example, produced by nucleation) can grow, percolate and form a network. This condition is satisfied for fundamental strings in the background of a kinating volume modulus rolling towards the asymptotic large volume region of moduli space. Such long kination epochs are motivated in string cosmology by both the electroweak hierarchy problem and the need to solve the overshoot problem. The tension of such a network today is set by the final vacuum; for phenomenologically appealing Large Volume Scenario (LVS) vacua, this would lead to a fundamental string network with $G \mu \sim 10^{-10}$., Comment: 6 pages

We show that for every non-spherical set $X$ in $\mathbb{E}^d$, there exists a natural number $m$ and a red/blue-colouring of $\mathbb{E}^n$ for every $n$ such that there is no red copy of X and no blue progression of length $m$ with each consecutive point at distance $1$. This verifies a conjecture of Wu and the first author.

This is a short account, based on a talk given at the 2024 Moriond Cosmology Conference, of where and why string theory matters in early universe cosmology. It is written for a cosmology audience predisposed to be at best sceptical, and at worst contemptuous, of the notion that either quantum gravity or string theory has any relevance for their discipline. I cover inflation, CMB tensor modes, extended kination or tracker epochs and reheating., Comment: 8 pages, based on talk at 2024 Moriond Cosmology Conference

In multi-parameter quantum metrology, the resource of entanglement can lead to an increase in efficiency of the estimation process. Entanglement can be used in the state preparation stage, or the measurement stage, or both, to harness this advantage; here we focus on the role of entangling measurements. Specifically, entangling or collective measurements over multiple identical copies of a probe state are known to be superior to measuring each probe individually, but the extent of this improvement is an open problem. It is also known that such entangling measurements, though resource-intensive, are required to attain the ultimate limits in multi-parameter quantum metrology and quantum information processing tasks. In this work we investigate the maximum precision improvement that collective quantum measurements can offer over individual measurements for estimating parameters of qudit states, calling this the 'collective quantum enhancement'. We show that, whereas the maximum enhancement can, in principle, be a factor of $n$ for estimating $n$ parameters, this bound is not tight for large $n$. Instead, our results prove an enhancement linear in dimension of the qudit is possible using collective measurements and lead us to conjecture that this is the maximum collective quantum enhancement in any local estimation scenario., Comment: 22 pages, 9 figures, 10 appendices; presented at AIP Summer Meeting 2023

We show that for every integer $k\geq3$, the set of Tur\'an densities of $k$-uniform hypergraphs has an accumulation point in $[0,1)$. In particular, $1/2$ is an accumulation point for the set of Tur\'an densities of $3$-uniform hypergraphs., Comment: 6 pages

Let $ES_{\ell}(n)$ be the minimum $N$ such that every $N$-element point set in the plane contains either $\ell$ collinear members or $n$ points in convex position. We prove that there is a constant $C>0$ such that, for each $\ell, n \ge 3$, $$ (3\ell - 1) \cdot 2^{n-5} < ES_{\ell}(n) < \ell^2 \cdot 2^{n+ C\sqrt{n\log n}}.$$ A similar extension of the well-known Erd\H os--Szekeres cups-caps theorem is also proved.

We demonstrate an atom interferometer measurement protocol compatible with operation on a dynamic platform. Our method employs two open interferometers, derived from the same atomic source, with different interrogation times to eliminate initial velocity dependence while retaining precision, accuracy, and long term stability. We validate the protocol by measuring gravitational tides, achieving a precision of 4.5 {\mu}Gal in 2000 runs (6.7 h), marking the first demonstration of inertial quantity measurement with open atom interferometry that achieves long-term phase stability.

A graph $H$ is said to be positive if the homomorphism density $t_H(G)$ is non-negative for all weighted graphs $G$. The positive graph conjecture proposes a characterisation of such graphs, saying that a graph is positive if and only if it is symmetric, in the sense that it is formed by gluing two copies of some subgraph along an independent set. We prove several results relating to this conjecture. First, we make progress towards the conjecture itself by showing that any connected positive graph must have a vertex of even degree. We then make use of this result to identify some new counterexamples to the analogue of Sidorenko's conjecture for hypergraphs. In particular, we show that, for $r$ odd, every $r$-uniform tight cycle is a counterexample, generalising a recent result of Conlon, Lee and Sidorenko that dealt with the case $r=3$. Finally, we relate the positive graph conjecture to the emerging study of graph codes by showing that any positive graph has vanishing graph code density, thereby improving a result of Alon who proved the same result for symmetric graphs. Our proofs make use of a variety of tools and techniques, including the properties of independence polynomials, hypergraph quasirandomness and discrete Fourier analysis., Comment: 14 pages

Answering a question of Erd\H{o}s and Graham, we show that for each fixed positive rational number $x$ the number of ways to write $x$ as a sum of reciprocals of distinct positive integers each at most $n$ is $2^{(c_x + o(1))n}$ for an explicit constant $c_x$ increasing with $x$., Comment: 8 pages

A fundamental problem in Ramsey theory is to determine the growth rate in terms of $n$ of the Ramsey number $r(H, K_n^{(3)})$ of a fixed $3$-uniform hypergraph $H$ versus the complete $3$-uniform hypergraph with $n$ vertices. We study this problem, proving two main results. First, we show that for a broad class of $H$, including links of odd cycles and tight cycles of length not divisible by three, $r(H, K_n^{(3)}) \ge 2^{\Omega_H(n \log n)}$. This significantly generalizes and simplifies an earlier construction of Fox and He which handled the case of links of odd cycles and is sharp both in this case and for all but finitely many tight cycles of length not divisible by three. Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph $H$ for which $r(H, K_n^{(3)})$ is superpolynomial in $n$. This provides the first example of a separation between $r(H,K_n^{(3)})$ and $r(H,K_{n,n,n}^{(3)})$, since the latter is known to be polynomial in $n$ when $H$ is linear., Comment: 22 pages

The symmetric logarithmic derivative Cram\'er-Rao bound (SLDCRB) provides a fundamental limit to the minimum variance with which a set of unknown parameters can be estimated in an unbiased manner. It is known that the SLDCRB can be saturated provided the optimal measurements for the individual parameters commute with one another. However, when this is not the case the SLDCRB cannot be attained in general. In the experimentally relevant setting, where quantum states are measured individually, necessary and sufficient conditions for when the SLDCRB can be saturated are not known. In this setting the SLDCRB is attainable provided the SLD operators can be chosen to commute on an extended Hilbert space. However, beyond this relatively little is known about when the SLD operators can be chosen in this manner. In this paper we present explicit examples which demonstrate novel aspects of this condition. Our examples demonstrate that the SLD operators commuting on any two of the following three spaces: support space, support-kernel space and kernel space, is neither a necessary nor sufficient condition for commutativity on the extended space. We present a simple analytic example showing that the Nagaoka-Hayashi Cram\'er-Rao bound is not always attainable. Finally, we provide necessary and sufficient conditions for the attainability of the SLDCRB in the case when the kernel space is one-dimensional. These results provide new information on the necessary and sufficient conditions for the attainability of the SLDCRB., Comment: Comments very welcome

In $1996$, H.-D. Cao constructed a $U(n)$-invariant steady gradient K\"ahler-Ricci soliton on $\mathbb{C}^{n}$ and asked whether every steady gradient K\"ahler-Ricci soliton of positive curvature on $\mathbb{C}^{n}$ is necessarily $U(n)$-invariant (and hence unique up to scaling). Recently, Apostolov-Cifarelli answered this question in the negative for $n=2$. Here, we construct a family of $U(1)\times U(n-1)$-invariant, but not $U(n)$-invariant, complete steady gradient K\"ahler-Ricci solitons with strictly positive curvature operator on real $(1,\,1)$-forms (in particular, with strictly positive sectional curvature) on $\mathbb{C}^{n}$ for $n\geq3$, thereby answering Cao's question in the negative for $n\geq3$. This family of steady Ricci solitons interpolates between Cao's $U(n)$-invariant steady K\"ahler-Ricci soliton and the product of the cigar soliton and Cao's $U(n-1)$-invariant steady K\"ahler-Ricci soliton. This provides the K\"ahler analog of the Riemannian flying wings construction of Lai. In the process of the proof, we also demonstrate that the almost diameter rigidity of $\mathbb{P}^{n}$ endowed with the Fubini-Study metric does not hold even if the curvature operator is bounded below by $2$ on real $(1,\,1)$-forms., Comment: Appendix B added; more references added

New light hidden sector degrees of freedom represent one of the most approaches to going beyond the Standard Model. I give a short account of how such WISP candidates naturally appear in string compactifications and some descriptions of ways that they can affect early universe cosmology., Comment: 6 pages, contribution to the Proceedings of the 1st General Meeting of the COSMIC WISPers COST Action CA21106

We perform a detailed study of stringy moduli-driven cosmologies between the end of inflation and the commencement of the Hot Big Bang, including both the background and cosmological perturbations: a period that can cover half the lifetime of the universe on a logarithmic scale. Compared to the standard cosmology, stringy cosmologies motivate extended kination, tracker and moduli-dominated epochs involving significantly trans-Planckian field excursions. Conventional effective field theory is unable to control Planck-suppressed operators and so such epochs require a stringy completion for a consistent analysis. Perturbation growth in these stringy cosmologies is substantially enhanced compared to conventional cosmological histories. The transPlanckian field evolution results in radical changes to Standard Model couplings during this history and we outline potential applications to baryogenesis, dark matter and gravitational wave production., Comment: v1:61 pages, 2 figures, comments welcome. v2: References added, typos corrected

Quantum mechanics offers the possibility of unconditionally secure communication between multiple remote parties. Security proofs for such protocols typically rely on bounding the capacity of the quantum channel in use. In a similar manner, Cram\'er-Rao bounds in quantum metrology place limits on how much information can be extracted from a given quantum state about some unknown parameters of interest. In this work we establish a connection between these two areas. We first demonstrate a three-party sensing protocol, where the attainable precision is dependent on how many parties work together. This protocol is then mapped to a secure access protocol, where only by working together can the parties gain access to some high security asset. Finally, we map the same task to a communication protocol where we demonstrate that a higher mutual information can be achieved when the parties work collaboratively compared to any party working in isolation., Comment: Close to published version: npj Quantum Inf 10, 35 (2024)

We show that for every positive integer $k$ there are positive constants $C$ and $c$ such that if $A$ is a subset of $\{1, 2, \dots, n\}$ of size at least $C n^{1/k}$, then, for some $d \leq k-1$, the set of subset sums of $A$ contains a homogeneous $d$-dimensional generalized arithmetic progression of size at least $c|A|^{d+1}$. This strengthens a result of Szemer\'edi and Vu, who proved a similar statement without the homogeneity condition. As an application, we make progress on the Erd\H{o}s--Straus non-averaging sets problem, showing that every subset $A$ of $\{1, 2, \dots, n\}$ of size at least $n^{\sqrt{2} - 1 + o(1)}$ contains an element which is the average of two or more other elements of $A$. This gives the first polynomial improvement on a result of Erd\H{o}s and S\'ark\"ozy from 1990., Comment: 34 pages

Determination of optimal control protocols for Majorana zero modes during their exchange is a crucial step towards the realisation of the topological quantum computer. In this paper, we study the finite-time exchange process of Majorana zero modes on a network formed by coupled $p$-wave superconducting one-dimensional nanowires. We provide scalable computational tools for optimising such an exchange process relying on deep learning techniques. To accomplish the scalability, we derive and implement an analytic formula for the gradient of the quantum infidelity which measures the error in the topological quantum gate generation in the Majorana zero modes exchange. Our optimisation strategy relies on learning the optimised transport protocol via a neural net which is followed by direct gradient descent fine tuning. The optimised exchange protocols in the super-adiabatic regime discover the fact that the Majorana zero modes must necessarily stop before crossing a junction point in the network. We explain that this is caused by fast changes in the energy gap of the system whenever one of the Majorana zero modes approaches a junction point. In particular, the energy gap exhibits oscillations followed by a sharp jump. We explain this phenomenon analytically in the regime where the Majorana zero modes are completely localised. Finally, we study how the disorder in the quantum nanowire affects the exchange protocols. This shows that understanding the disorder pattern would allow one to improve quantum gate fidelity by one to two orders of magnitude., Comment: 16 pages

A simplicial complex $H$ consists of a pair of sets $(V,E)$ where $V$ is a set of vertices and $E\subseteq\mathscr{P}(V)$ is a collection of subsets of $V$ closed under taking subsets. Given a simplicial complex $F$ and $n\in \mathbb N$, the extremal number $\text{ex}(n,F)$ is the maximum number of edges that a simplicial complex on $n$ vertices can have without containing a copy of $F$. We initiate the systematic study of extremal numbers in this context by asymptotically determining the extremal numbers of several natural simplicial complexes. In particular, we asymptotically determine the extremal number of a simplicial complex for which the extremal example has more than one incomplete layer., Comment: 22 pages

Quantum mechanics has withstood every experimental test thus far. However, it relies on ad-hoc postulates which require experimental verification. Over the past decade there has been a great deal of research testing these postulates, with numerous tests of Born's rule for determining probabilities and the complex nature of the Hilbert space being carried out. Although these tests are yet to reveal any significant deviation from textbook quantum theory, it remains important to conduct such tests in different configurations and using different quantum states. Here we perform the first such test using coherent states of light in a three-arm interferometer combined with homodyne detection. Our proposed configuration requires additional assumptions, but allows us to use quantum states which exist in a larger Hilbert space compared to previous tests. For testing Born's rule, we find that the third order interference is bounded to be $\kappa$ = 0.002 $\pm$ 0.004 and for testing whether quantum mechanics is complex or not we find a Peres parameter of F = 1.0000 $\pm$ 0.0003 (F = 1 corresponds to the expected complex quantum mechanics). We also use our experiment to test Glauber's theory of optical coherence., Comment: 11 pages, comments very welcome

We construct many new examples of complete Calabi-Yau metrics of maximal volume growth on certain smoothings of Cartesian products of Calabi-Yau cones with smooth cross-sections. A detailed description of the geometry at infinity of these metrics is given in terms of a compactification by a manifold with corners obtained through the notion of weighted blow-up for manifolds with corners. A key analytical step in the construction of these Calabi-Yau metrics is to derive good mapping properties of the Laplacian on some suitable weighted H\"older spaces. Our methods also produce singular Calabi-Yau metrics with an isolated conical singularity modelled on a Calabi-Yau cone distinct from the tangent cone at infinity, in particular yielding a transition behavior between different Calabi-Yau cones as conjectured by Yang Li. This is used to exhibit many examples where the tangent cone at infinity does not uniquely specify a complete Calabi-Yau metric with exact K\"ahler form., Comment: 64 pages, added the construction of singular Calabi-Yau metrics with an isolated conical singularity modelled on a Calabi-Yau cone distinct from the tangent cone at infinity, thus interpolating between two different Calabi-Yau cones

An old problem in discrete geometry, originating with Kupitz, asks whether there is a fixed natural number $k$ such that every finite set of points in the plane has a line through at least two of its points where the number of points on either side of this line differ by at most $k$. We give a negative answer to a natural variant of this problem, showing that for every natural number $k$ there exists a finite set of points in the plane together with a pseudoline arrangement such that each pseudoline contains at least two points and there is a pseudoline through any pair of points where the number of points on either side of each pseudoline differ by at least $k$., Comment: 27 pages, 24 figures