arXiv subjects

Christian Winter

Publications and source records attributed to Christian Winter.

18 recordsLinked to original sources

arXiv2024

Ramsey numbers for partially ordered sets

In this thesis, we present quantitative Ramsey-type results in the setting of finite sets that are equipped with a partial order, so-called posets. A prominent example of a poset is the Boolean lattice $Q_n$, which consists of all subsets of $\{1,\dots,n\}$, ordered by inclusion. For posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the smallest $N$ such that no matter how the elements of $Q_N$ are colored in blue and red, there is either an induced subposet isomorphic to $P$ in which every element is colored blue, or an induced subposet isomorphic to $Q$ in which every element is colored red. The central focus of this thesis is to investigate $R(P,Q_n)$, where $P$ is fixed and $n$ grows large. Our results contribute to an active area of discrete mathematics, which studies the existence of large homogeneous substructures in host structures with local constraints, introduced for graphs by Erd\H{o}s and Hajnal. We provide an asymptotically tight bound on $R(P,Q_n)$ for $P$ from several classes of posets, and show a dichotomy in the asymptotic behavior of $R(P,Q_n)$, depending on whether $P$ contains a subposet isomorphic to one of two specific posets. A fundamental question in the study of poset Ramsey numbers is to determine the asymptotic behavior of $R(Q_n,Q_n)$ for large $n$. In this dissertation, we present improvements on the known lower and upper bound on $R(Q_n,Q_n)$. Moreover, we explore variations of the poset Ramsey setting, including Erd\H{o}s-Hajnal-type questions when the small forbidden poset has a non-monochromatic color pattern, and so-called weak poset Ramsey numbers, which are concerned with non-induced subposets.

math.CO↗

arXiv2024

Diagonal poset Ramsey numbers

A poset $(Q,\le_Q)$ contains an induced copy of a poset $(P,\le_P)$ if there exists an injective mapping $\phi\colon P\to Q$ such that for any two elements $X,Y\in P$, $X\le_P Y$ if and only if $\phi(X)\le_Q \phi(Y)$. By $Q_n$ we denote the Boolean lattice $(2^{[n]},\subseteq)$. The poset Ramsey number $R(P,Q)$ for posets $P$ and $Q$ is the least integer $N$ for which any coloring of the elements of $Q_N$ in blue and red contains either a blue induced copy of $P$ or a red induced copy of $Q$. In this paper, we show that $R(Q_m,Q_n)\le nm-\big(1-o(1)\big)n\log m$ where $n\ge m$ and $m$ is sufficiently large. This improves the best known upper bound on $R(Q_n,Q_n)$ from $n^2-n+2$ to $n^2-\big(1-o(1)\big) n\log n$. Furthermore, we determine $R(P,P)$ where $P$ is an $n$-fork or $n$-diamond up to an additive constant of $2$. A poset $(Q,\le_Q)$ contains a weak copy of $(P,\le_P)$ if there is an injection $\psi\colon P\to Q$ such that $\psi(X)\le_Q \psi(Y)$ for any $X,Y\in P$ with $X\le_P Y$. The weak poset Ramsey number $R^{\text{w}}(P,Q)$ is the smallest $N$ for which any blue/red-coloring of $Q_N$ contains a blue weak copy of $P$ or a red weak copy of $Q$. We show that $R^{\text{w}}(Q_n,Q_n)\le 0.96n^2$.

math.CO↗

arXiv2023

Erd\H{o}s-Hajnal problems for posets

We say that a poset $(Q,\le_{Q})$ contains an induced copy of a poset $(P,\le_P)$ if there is an injective function $\phi\colon P\to Q$ such that for every two $X,Y\in P$,\;\;$X\le_P Y$ if and only if $\phi(X)\le_Q \phi(Y)$. We denote the Boolean lattice $(2^{[n]},\subseteq)$ by $Q_n$. Given a fixed $2$-coloring $c$ of a poset $P$, the poset Erd\H{o}s-Hajnal number of this colored poset is the smallest integer $N$ such that every $2$-coloring of the Boolean lattice $Q_N$ contains an induced copy of $P$ colored as in $c$, or a monochromatic induced copy of $Q_n$. We present bounds on the poset Erd\H{o}s-Hajnal number of general colored posets, antichains, chains, and small Boolean lattices. Let the poset Ramsey number $R(Q_n,Q_n)$ be the least $N$ such that every $2$-coloring of $Q_N$ contains a monochromatic induced copy of $Q_n$. As a corollary, we show that $R(Q_n,Q_n)> 2.02n$, improving on the best known lower bound $2n+1$ by Cox and Stolee \cite{CS}.

math.CO↗

arXiv2023

A note on asymmetric hypergraphs

A $k$-graph $\mathcal{G}$ is asymmetric if there does not exist an automorphism on $\mathcal{G}$ other than the identity, and $\mathcal{G}$ is called minimal asymmetric if it is asymmetric but every non-trivial induced sub-hypergraph of $\mathcal{G}$ is non-asymmetric. Extending a result of Jiang and Ne\v{s}et\v{r}il, we show that for every $k$-graph, $k\ge3$, there exist infinitely many minimal asymmetric $k$-graphs which have maximum degree $2$ and are linear. Further, we show that there are infinitely many $2$-regular asymmetric $k$-graphs for $k\ge3$.

math.CO↗

arXiv2023

On graphs embeddable in a layer of a hypercube and their extremal numbers

A graph is cubical if it is a subgraph of a hypercube. For a cubical graph $H$ and a hypercube $Q_n$, $ex(Q_n, H)$ is the largest number of edges in an $H$-free subgraph of $Q_n$. If $ex(Q_n, H)$ is equal to a positive proportion of the number of edges in $Q_n$, $H$ is said to have positive Tur\'an density in a hypercube; otherwise it has zero Tur\'an density. Determining $ex(Q_n, H)$ and even identifying whether $H$ has positive or zero Tur\'an density remains a widely open question for general $H$. In this paper we focus on layered graphs, i.e., graphs that are contained in an edge-layer of some hypercube. Graphs $H$ that are not layered have positive Tur\'an density because one can form an $H$-free subgraph of $Q_n$ consisting of edges of every other layer. For example, a $4$-cycle is not layered and has positive Tur\'an density. However, in general it is not obvious what properties layered graphs have. We give a characterisation of layered graphs in terms of edge-colorings. We show that most non-trivial subdivisions have zero Tur\'an density, extending known results on zero Tur\'an density of even cycles of length at least $12$ and of length $8$. However, we prove that there are cubical graphs of girth $8$ that are not layered and thus having positive Tur\'an density. The cycle of length $10$ remains the only cycle for which it is not known whether its Tur\'an density is positive or not. We prove that $ex(Q_n, C_{10})= \Omega(n2^n/ \log^a n)$, for a constant $a$, showing that the extremal number for a $10$-cycle behaves differently from any other cycle of zero Tur\'an density.

math.CO↗

arXiv2023

Poset Ramsey number $R(P,Q_n)$. III. Chain Compositions and Antichains

An induced subposet $(P_2,\le_2)$ of a poset $(P_1,\le_1)$ is a subset of $P_1$ such that for every two $X,Y\in P_2$, $X\le_2 Y$ if and only if $X\le_1 Y$. The Boolean lattice $Q_n$ of dimension $n$ is the poset consisting of all subsets of $\{1,\dots,n\}$ ordered by inclusion. Given two posets $P_1$ and $P_2$ the poset Ramsey number $R(P_1,P_2)$ is the smallest integer $N$ such that in any blue/red coloring of the elements of $Q_N$ there is either a monochromatically blue induced subposet isomorphic to $P_1$ or a monochromatically red induced subposet isomorphic to $P_2$. We provide upper bounds on $R(P,Q_n)$ for two classes of $P$: parallel compositions of chains, i.e.\ posets consisting of disjoint chains which are pairwise element-wise incomparable, as well as subdivided $Q_2$, which are posets obtained from two parallel chains by adding a common minimal and a common maximal element. This completes the determination of $R(P,Q_n)$ for posets $P$ with at most $4$ elements. If $P$ is an antichain $A_t$ on $t$ elements, we show that $R(A_t,Q_n)=n+3$ for $3\le t\le \log \log n$. Additionally, we briefly survey proof techniques in the poset Ramsey setting $P$ versus $Q_n$.

math.CO↗

arXiv2022

Poset Ramsey number $R(P,Q_n)$. II. N-shaped poset

Given partially ordered sets (posets) $(P, \leq_P)$ and $(P', \leq_{P'})$, we say that $P'$ contains a copy of $P$ if for some injective function $f\colon P\rightarrow P'$ and for any $A, B\in P$, $A\leq _P B$ if and only if $f(A)\leq_{P'} f(B)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that no matter how the elements of an $N$-dimensional Boolean lattice are colored in blue and red, there is either a copy of $P$ with all blue elements or a copy of $Q$ with all red elements. We focus on the poset Ramsey number $R(P, Q_n)$ for a fixed poset $P$ and an $n$-dimensional Boolean lattice $Q_n$, as $n$ grows large. It is known that $n+c_1(P) \leq R(P,Q_n) \leq c_2(P) n$, for positive constants $c_1$ and $c_2$. However, there is no poset $P$ known, for which $R(P, Q_n)> (1+\epsilon)n$, for $\epsilon >0$. This paper is devoted to a new method for finding upper bounds on $R(P, Q_n)$ using a duality between copies of $Q_n$ and sets of elements that cover them, referred to as blockers. We prove several properties of blockers and their direct relation to the Ramsey numbers. Using these properties we show that $R(\mathcal{N},Q_n)=n+\Theta(n/\log n)$, for a poset $\mathcal{N}$ with four elements $A, B, C, $ and $D$, such that $A<C$, $B<D$, $B<C$, and the remaining pairs of elements are incomparable.

math.CO↗

arXiv2022

Poset Ramsey Number $R(P,Q_n)$. II. Antichains

For two posets $(P,\le_P)$ and $(P',\le_{P'})$, we say that $P'$ contains a copy of $P$ if there exists an injective function $f\colon P'\to P$ such that for every two $X,Y\in P$, $X\le_P Y$ if and only if $f(X)\le_{P'} f(Y)$. Given two posets $P$ and $Q$, let the poset Ramsey number $R(P,Q)$ be the smallest integer $N$ such that any coloring of the elements of an $N$-dimensional Boolean lattice in blue or red contains either a copy of $P$ where all elements are blue or a copy of $Q$ where all elements are red. We determine the poset Ramsey number $R(A_t,Q_n)$ of an antichain versus a Boolean lattice for small $t$ by showing that $R(A_t,Q_n)=n+3$ for $3\le t\le \log \log n$.

math.CO↗

arXiv2022

Poset Ramsey number $R(P,Q_n)$. I. Complete multipartite posets

A poset $(P',\le_{P'})$ contains a copy of some other poset $(P,\le_P)$ if there is an injection $f\colon P'\to P$ where for every $X,Y\in P$, $X\le_P Y$ if and only if $f(X)\le_{P'} f(Y)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the smallest integer $N$ such that any blue/red coloring of a Boolean lattice of dimension $N$ contains either a copy of $P$ with all elements blue or a copy of $Q$ with all elements red. We denote by $K_{t_1,\dots,t_\ell}$ a complete $\ell$-partite poset, i.e.\ a poset consisting of $\ell$ pairwise disjoint sets $A^i$ of size $t_i$, $1\le i\le \ell$, such that for any $i,j\in\{1,\dots,\ell\}$ and any two $X\in A^{i}$ and $Y\in A^{j}$, $X<Y$ if and only if $i<j$. In this paper we show that $R(K_{t_1,\dots,t_\ell},Q_n)\le n+\frac{(2+o_n(1))\ell n}{\log n}$.

math.CO↗

arXiv2021

Poset Ramsey numbers: large Boolean lattice versus a fixed poset

Given partially ordered sets (posets) $(P, \leq_P)$ and $(P', \leq_{P'})$, we say that $P'$ contains a copy of $P$ if for some injective function $f: P\rightarrow P'$ and for any $X, Y\in P$, $X\leq _P Y$ if and only of $f(X)\leq_{P'} f(Y)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that no matter how the elements of an $N$-dimensional Boolean lattice are colored in blue and red, there is either a copy of $P$ with all blue elements or a copy of $Q$ with all red elements. We focus on a poset Ramsey number $R(P, Q_n)$ for a fixed poset $P$ and an $n$-dimensional Boolean lattice $Q_n$, as $n$ grows large. We show a sharp jump in behaviour of this number as a function of $n$ depending on whether or not $P$ contains a copy of either a poset $V$, i.e. a poset on elements $A, B, C$ such that $B>C$, $A>C$, and $A$ and $B$ incomparable, or a poset $\Lambda$, its symmetric counterpart. Specifically, we prove that if $P$ contains a copy of $V$ or $\Lambda$ then $R(P, Q_n) \geq n +\frac{1}{15} \frac{n}{\log n}$. Otherwise $R(P, Q_n) \leq n + c(P)$ for a constant $c(P)$. This gives the first non-marginal improvement of a lower bound on poset Ramsey numbers and as a consequence gives $R(Q_2, Q_n) = n + \Theta (\frac{n}{\log n})$.

math.CO↗

arXiv2021

Lower Bound on the Size-Ramsey Number of Tight Paths

The size-Ramsey number $R^{(k)}(H)$ of a $k$-uniform hypergraph $H$ is the minimum number of edges in a $k$-uniform hypergraph $G$ with the property that every `$2$-edge coloring' of $G$ contains a monochromatic copy of $H$. For $k\ge2$ and $n\in\mathbb{N}$, a $k$-uniform tight path on $n$ vertices $P^{(k)}_{n}$ is defined as a $k$-uniform hypergraph on $n$ vertices for which there is an ordering of its vertices such that the edges are all sets of $k$ consecutive vertices with respect to this order. We prove a lower bound on the size-Ramsey number of $k$-uniform tight paths, which is, considered assymptotically in both the uniformity $k$ and the number of vertices $n$, $R^{(k)}(P^{(k)}_{n})= \Omega\big(\log (k)n\big)$.

math.CO↗

arXiv2019

A unified model of ripples and dunes in water and planetary environments

Subaqueous and aeolian bedforms are ubiquitous on Earth and other planetary environments. However, it is still unclear which hydrodynamical mechanisms lead to the observed variety of morphologies of self-organized natural patterns such as ripples, dunes or compound bedforms. Here we present simulations with a coupled hydrodynamic and sediment transport model that resolve the initial and mature stages of subaqueous and aeolian bedform evolution in the limit of large flow thickness. We identify two types of bedforms consistent with subaqueous ripples and dunes, and separated by a gap in wavelength. This gap is explained in terms of an anomalous hydrodynamic response in the structure of the inner boundary layer that leads to a shift of the position of the maximum shear stress from upstream to downstream of the crest. This anomaly gradually disappears when the bed becomes hydrodynamically rough. By also considering the effect of the spatial relaxation of sediment transport we provide a new unifying framework to compare ripples and dunes in planetary environments to their terrestrial counterparts.

physics.geo-ph↗

arXiv2019

GeoBlocks: A Query-Cache Accelerated Data Structure for Spatial Aggregation over Polygons

As individual traffic and public transport in cities are changing, city authorities need to analyze urban geospatial data to improve transportation and infrastructure. To that end, they highly rely on spatial aggregation queries that extract summarized information from point data (e.g., Uber rides) contained in a given polygonal region (e.g., a city neighborhood). To support such queries, current analysis tools either allow only predefined aggregates on predefined regions and are thus unsuitable for exploratory analyses, or access the raw data to compute aggregate results on-the-fly, which severely limits the interactivity. At the same time, existing pre-aggregation techniques are inadequate since they maintain aggregates over rectangular regions. As a result, when applied over arbitrary polygonal regions, they induce an approximation error that cannot be bounded. In this paper, we introduce GeoBlocks, a novel pre-aggregating data structure that supports spatial aggregation over arbitrary polygons. GeoBlocks closely approximate polygons using a set of fine-grained grid cells and, in contrast to prior work, allow to bound the approximation error by adjusting the cell size. Furthermore, GeoBlocks employ a trie-like cache that caches aggregate results of frequently queried regions, thereby dynamically adapting to the skew inherently present in query workloads and improving performance over time. In summary, GeoBlocks outperform on-the-fly aggregation by up to three orders of magnitude, achieving the sub-second query latencies required for interactive exploratory analytics.

cs.DB↗

arXiv2019

Assessing the Applicability of Authorship Verification Methods

Authorship verification (AV) is a research subject in the field of digital text forensics that concerns itself with the question, whether two documents have been written by the same person. During the past two decades, an increasing number of proposed AV approaches can be observed. However, a closer look at the respective studies reveals that the underlying characteristics of these methods are rarely addressed, which raises doubts regarding their applicability in real forensic settings. The objective of this paper is to fill this gap by proposing clear criteria and properties that aim to improve the characterization of existing and future AV approaches. Based on these properties, we conduct three experiments using 12 existing AV approaches, including the current state of the art. The examined methods were trained, optimized and evaluated on three self-compiled corpora, where each corpus focuses on a different aspect of applicability. Our results indicate that part of the methods are able to cope with very challenging verification cases such as 250 characters long informal chat conversations (72.7% accuracy) or cases in which two scientific documents were written at different times with an average difference of 15.6 years (> 75% accuracy). However, we also identified that all involved methods are prone to cross-topic verification cases.

cs.LG↗

arXiv2018

Unary and Binary Classification Approaches and their Implications for Authorship Verification

Retrieving indexed documents, not by their topical content but their writing style opens the door for a number of applications in information retrieval (IR). One application is to retrieve textual content of a certain author X, where the queried IR system is provided beforehand with a set of reference texts of X. Authorship verification (AV), which is a research subject in the field of digital text forensics, is suitable for this purpose. The task of AV is to determine if two documents (i.e. an indexed and a reference document) have been written by the same author X. Even though AV represents a unary classification problem, a number of existing approaches consider it as a binary classification task. However, the underlying classification model of an AV method has a number of serious implications regarding its prerequisites, evaluability, and applicability. In our comprehensive literature review, we observed several misunderstandings regarding the differentiation of unary and binary AV approaches that require consideration. The objective of this paper is, therefore, to clarify these by proposing clear criteria and new properties that aim to improve the characterization of existing and future AV approaches. Given both, we investigate the applicability of eleven existing unary and binary AV methods as well as four generic unary classification algorithms on two self-compiled corpora. Furthermore, we highlight an important issue concerning the evaluation of AV methods based on fixed decision criterions, which has not been paid attention in previous AV studies.

cs.IR↗

arXiv2018

A note on saturation for Berge-G hypergraphs

For a graph G, a hypergraph H is called Berge-G if there is a hypergraph H', isomorphic to H, containing all vertices of G, so that e is contained in f(e) for each edge e of G, where f is a bijection between E(G) and E(H'). The set of all Berge-G hypergraphs is denoted B(G). A hypergraph H is called Berge-G saturated if it does not contain any subhypergraph from B(G), but adding any new hyperedge of size at least 2 to H creates such a subhypergraph. Each Berge-G saturated hypergraph has at least |E(G)|-1 hyperedges. We show that for each graph G that is not a certain star and for any n at least |V(G)|, there is a Berge-G saturated hypergraph on n vertices and exactly |E(G)|-1 hyperedges. This solves a problem of finding a saturated hypergraph on n vertices with the smallest number of edges exactly.

math.CO↗

arXiv2017

Authorship Verification based on Compression-Models

Compression models represent an interesting approach for different classification tasks and have been used widely across many research fields. We adapt compression models to the field of authorship verification (AV), a branch of digital text forensics. The task in AV is to verify if a questioned document and a reference document of a known author are written by the same person. We propose an intrinsic AV method, which yields competitive results compared to a number of current state-of-the-art approaches, based on support vector machines or neural networks. However, in contrast to these approaches our method does not make use of machine learning algorithms, natural language processing techniques, feature engineering, hyperparameter optimization or external documents (a common strategy to transform AV from a one-class to a multi-class classification problem). Instead, the only three key components of our method are a compressing algorithm, a dissimilarity measure and a threshold, needed to accept or reject the authorship of the questioned document. Due to its compactness, our method performs very fast and can be reimplemented with minimal effort. In addition, the method can handle complicated AV cases where both, the questioned and the reference document, are not related to each other in terms of topic or genre. We evaluated our approach against publicly available datasets, which were used in three international AV competitions. Furthermore, we constructed our own corpora, where we evaluated our method against state-of-the-art approaches and achieved, in both cases, promising results.

cs.IR↗

arXiv2016

Aberration-free volumetric high-speed imaging of in vivo retina

Research and medicine rely on non-invasive optical techniques to image living tissue with high resolution in space and time. But so far a single data acquisition could not provide entirely diffraction-limited tomographic volumes of rapidly moving or changing targets, which additionally becomes increasingly difficult in the presence of aberrations, e.g., when imaging retina in vivo. We show, that a simple interferometric setup based on parallelized optical coherence tomography acquires volumetric data with 10 billion voxels per second, exceeding previous imaging speeds by an order of magnitude. This allows us to computationally obtain and correct defocus and aberrations resulting in entirely diffraction-limited volumes. As demonstration, we imaged living human retina with clearly visible nerve fiber layer, small capillary networks, and photoreceptor cells, but the technique is also applicable to obtain phase-sensitive volumes of other scattering structures at unprecedented acquisition speeds.

physics.optics↗