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We study the number of queries needed to identify a monotone Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$. A query consists of a 0-1-sequence, and the answer is the value of $f$ on that sequence. It is well-known that the number of queries needed is $\binom{n}{\lfloor n/2\rfloor}+\binom{n}{\lfloor n/2\rfloor+1}$ in general. Here we study a variant where $f$ has $k$ ``reasons'' to be 1, i.e., its disjunctive normal form has $k$ conjunctions if the redundant conjunctions are deleted. This problem is equivalent to identifying an upfamily in $2^{[n]}$ that has exactly $k$ minimal members. We find the asymptotics on the number of queries needed for fixed $k$. We also study the non-adaptive version of the problem, where the queries are asked at the same time, and determine the exact number of queries for most values of $k$ and $n$.

Recent progress in quantum machine learning has sparked interest in using quantum methods to tackle classical control problems via quantum reinforcement learning. However, the classical reinforcement learning environments often scale to high dimensional problem spaces, which represents a challenge for the limited and costly resources available for quantum agent implementations. We propose to solve this dimensionality challenge by a classical autoencoder and a quantum agent together, where a compressed representation of observations is jointly learned in a hybrid training loop. The latent representation of such an autoencoder will serve as a tailored observation space best suited for both the control problem and the QPU architecture, aligning with the agent's requirements. A series of numerical experiments are designed for a performance analysis of the latent-space learning method. Results are presented for different control problems and for both photonic (continuous-variable) and qubit-based agents, to show how the QNN learning process is improved by the joint training., Comment: 20 pages, 7 figures

This paper investigates photonic continuous-variable Born machines (CVBMs), which utilize photonic quantum states as resources for continuous probability distributions. Implementing exact gradient descent in the CVBM training process is often infeasible, bringing forward the need to approximate the gradients using an estimator obtained from a smaller number of samples, obtaining a quantum stochastic gradient descent (SGD) method. In this work, the ability to train CVBMs is analyzed using stochastic gradients obtained using relatively few samples from the probability distribution corresponding to homodyne measurement. The main obstacle to this analysis is that classically simulating CVBMs and obtaining samples is a demanding task, while a large number of iterations are needed to achieve convergence. The present research is enabled by a novel strategy to simulate homodyne detections of generic multimode photonic states using a classical computer. With this approach, a more comprehensive study of CVBMs is made possible, and the training of multimode CVBMs is demonstrated with parametric quantum circuits considerably larger than in previous articles. More specifically, we use the proposed algorithm to demonstrate learning of multimode quantum distributions using CVBMs. Moreover, successful CVBM trainings were demonstrated with the use of stochastic gradients., Comment: 7 pages, 8 figures

Let $M_k$ be a $2n$-vertex graph with $n$ pairwise disjoint edges and let $\mathcal{H}^{(p,s)}(n)$ be the family of subsets of $V(M_n)$ that span exactly $p$ edges and $s$ isolated vertices. We prove that for $n\ge 2p+s$ this family has the Erd\H{o}s--Ko--Rado property: the size of the largest intersecting family equals to the number of sets containing a fixed vertex. The bound $n\ge 2p+s$ is the best possible, improving a recent theorem with $n\ge 2p+2s$ by Fuentes and Kamat., Comment: 4 pages

This paper presents a comprehensive study on the possible hybrid quantum-classical autoencoder architectures for end-to-end radio communication against noisy channel conditions using standard encoded radio signals. The hybrid scenarios include single-sided, i.e., quantum encoder (transmitter) or quantum decoder (receiver), as well as fully quantum channel autoencoder (transmitter-receiver) systems. We provide detailed formulas for each scenario and validate our model through an extensive set of simulations. Our results demonstrate model robustness and adaptability. Supporting experiments are conducted utilizing 4-QAM and 16-QAM schemes and we expect that the model is adaptable to more general encoding schemes. We explore model performance against both additive white Gaussian noise and Rayleigh fading models. Our findings highlight the importance of designing efficient quantum neural network architectures for meeting application performance constraints -- including data re-uploading methods, encoding schemes, and core layer structures. By offering a general framework, this work paves the way for further exploration and development of quantum machine learning applications in radio communication., Comment: 12 pages, 6 figures

Leveraging the intrinsic probabilistic nature of quantum systems, generative quantum machine learning (QML) offers the potential to outperform classical learning models. Current generative QML algorithms mostly rely on general-purpose models that, while being very expressive, face several training challenges. A potential way to address these setbacks involves constructing problem-informed models capable of more efficient training on structured problems. In particular, probabilistic graphical models provide a flexible framework for representing structure in generative learning problems and can thus be exploited to incorporate inductive bias in QML algorithms. In this work, we propose a problem-informed quantum circuit Born machine Ansatz for learning the joint probability distribution of random variables, with independence relations efficiently represented by a Markov network (MN). We further demonstrate the applicability of the MN framework in constructing generative learning benchmarks and compare our model's performance to previous designs, showing it outperforms problem-agnostic circuits. Based on a preliminary analysis of trainability, we narrow down the class of MNs to those exhibiting favorable trainability properties. Finally, we discuss the potential of our model to offer quantum advantage in the context of generative learning., Comment: 11+5 pages, 10 figures

We introduce the Piquasso quantum programming framework, a full-stack open-source software platform for the simulation and programming of photonic quantum computers. Piquasso can be programmed via a high-level Python programming interface enabling users to perform efficient quantum computing with discrete and continuous variables. Via optional high-performance C++ backends, Piquasso provides state-of-the-art performance in the simulation of photonic quantum computers. The Piquasso framework is supported by an intuitive web-based graphical user interface where the users can design quantum circuits, run computations, and visualize the results., Comment: The source code is made available at https://github.com/Budapest-Quantum-Computing-Group/piquasso, and the documentation is published at https://piquasso.readthedocs.io/; November 2024 update: code examples adapted to Piquasso v5.0.0, presentation improved, minor errors corrected, references updated, new "Benchmarks" section added

We study the query complexity on slices of Boolean functions. Among other results we show that there exists a Boolean function for which we need to query all but 7 input bits to compute its value, even if we know beforehand that the number of 0's and 1's in the input are the same, i.e., when our input is from the middle slice. This answers a question of Byramji. Our proof is non-constructive, but we also propose a concrete candidate function that might have the above property. Our results are related to certain natural discrepancy type questions that, somewhat surprisingly, have not been studied before., Comment: 11 pages

Quantum neural networks are emerging as potential candidates to leverage noisy quantum processing units for applications. Here we introduce hybrid quantum-classical autoencoders for end-to-end radio communication. In the physical layer of classical wireless systems, we study the performance of simulated architectures for standard encoded radio signals over a noisy channel. We implement a hybrid model, where a quantum decoder in the receiver works with a classical encoder in the transmitter part. Besides learning a latent space representation of the input symbols with good robustness against signal degradation, a generalized data re-uploading scheme for the qubit-based circuits allows to meet inference-time constraints of the application., Comment: 6 pages, 8 figures

We study $\mathrm{exa}_k(n,F)$, the largest number of edges in an $n$-vertex graph $G$ that contains exactly $k$ copies of a given subgraph $F$. The case $k=0$ is the Tur\'an number $\mathrm{ex}(n,F)$ that is among the most studied parameters in extremal graph theory. We show that for any $F$ and $k$, $\mathrm{exa}_k(n,F)=(1+o(1))\mathrm{ex}(n,F))$ and determine the exact values of $\mathrm{exa}_k(n,K_3)$ and $\mathrm{exa}_1(n,K_r)$ for $n$ large enough. We also explore a connection to the following well-known problem in search theory. We are given a graph of order $n$ that consists of an unknown copy of $F$ and some isolated vertices. We can ask pairs of vertices as queries, and the answer tells us whether there is an edge between those vertices. Our goal is to describe the graph using as few queries as possible. Aigner and Triesch in 1990 showed that the number of queries needed is at least $\binom{n}{2}-\mathrm{exa}_1(n,F)$. Among other results we show that the number of queries that were answered NO is at least $\binom{n}{2}-\mathrm{exa}_1(n,F)$., Comment: 15 pages

The classical No-Three-In-Line problem seeks the maximum number of points that may be selected from an $n\times n$ grid while avoiding a collinear triple. The maximum is well known to be linear in $n$. Following a question of Erde, we seek to select sets of large density from the infinite grid $Z^{2}$ while avoiding a collinear triple. We show the existence of such a set which contains $\Theta(n/\log^{1+\varepsilon}n)$ points in $[1,n]^{2}$ for all $n$, where $\varepsilon>0$ is an arbitrarily small real number. We also give computational evidence suggesting that a set of lattice points may exist that has at least $n/2$ points on every large enough $n\times n$ grid., Comment: 12 pages, 3 figures

We determine normal forms of the multiplication of four-dimensional anti-commutative algebras over a field $\mathbb K$ of characteristic zero having an analogous family of flags of subalgebras as the four-dimensional non-Lie binary Lie algebras, and hence can be considered as the closest relatives of binary Lie algebras. These algebras are extensions of $\mathbb K$ by the 3-dimensional nilpotent Lie algebra and at the same time extensions of a two-dimensional Lie algebra by a two-dimensional abelian algebra. We describe their groups of automorphisms as extensions of a subgroup of the group of automorphisms of the three-dimensional nilpotent Lie algebra by $\mathbb K$., Comment: 22 pages

In extremal set theory our usual goal is to find the maximal size of a family of subsets of an $n$-element set satisfying a condition. A condition is called chain-dependent, if it is satisfied for a family if and only if it is satisfied for its intersections with the $n!$ full chains. We introduce a method to handle problems with such conditions, then show how it can be used to prove three classic theorems. Then, a theorem about families containing no two sets such that $A\subset B$ and $\lambda \cdot |A| \le |B|$ is proved. Finally, we investigate problems where instead of the size of the family, the number of $\ell$-chains is maximized. Our method is to define a weight function on the sets (or $\ell$-chains) and use it in a double counting argument involving full chains., Comment: 10 pages

We prove the following the generalized Tur\'an type result. A collection $\mathcal{T}$ of $r$ sets is an $r$-triangle if for every $T_1,T_2,\dots,T_{r-1}\in \mathcal{T}$ we have $\cap_{i=1}^{r-1}T_i\neq\emptyset$, but $\cap_{T\in \mathcal{T}}T$ is empty. A family $\mathcal{F}$ of sets is $r$-wise intersecting if for any $F_1,F_2,\dots,F_r\in \mathcal{F}$ we have $\cap_{i=1}^rF_i\neq \emptyset$ or equivalently if $\mathcal{F}$ does not contain any $m$-triangle for $m=2,3,\dots,r$. We prove that if $n\ge n_0(r,k)$, then the $r$-wise intersecting family $\mathcal{F}\subseteq \binom{[n]}{k}$ containing the most number of $(r+1)$-triangles is isomorphic to $\{F\in \binom{[n]}{k}:|F\cap [r+1]|\ge r\}$., Comment: 7 pages

For given posets $P$ and $Q$ and an integer $n$, the generalized Tur\'an problem for posets, asks for the maximum number of copies of $Q$ in a $P$-free subset of the $n$-dimensional Boolean lattice, $2^{[n]}$. In this paper, among other results, we show the following: (i) For every $n\geq 5$, the maximum number of $2$-chains in a butterfly-free subfamily of $2^{[n]}$ is $\left\lceil\frac{n}{2}\right\rceil\binom{n}{\lfloor n/2\rfloor}$. (ii) For every fixed $s$, $t$ and $k$, a $K_{s,t}$-free family in $2^{[n]}$ has $O\left(n\binom{n}{\lfloor n/2\rfloor}\right)$ $k$-chains. (iii) For every $n\geq 3$, the maximum number of $2$-chains in an $\textbf{N}$-free family is $\binom{n}{\lfloor n/2\rfloor}$, where $\textbf{N}$ is a poset on 4 distinct elements $\{p_1,p_2,q_1,q_2\}$ for which $p_1 < q_1$, $p_2 < q_1$ and $p_2 < q_2$. (iv) We also prove exact results for the maximum number of $2$-chains in a family that has no $5$-path and asymptotic estimates for the number of $2$-chains in a family with no $6$-path., Comment: 13 pages, 3 figures

For posets $P$ and $Q$, extremal and saturation problems about weak and strong $P$-free subposets of $Q$ have been studied mostly in the case $Q$ is the Boolean poset $Q_n$, the poset of all subsets of an $n$-element set ordered by inclusion. In this paper, we study some instances of the problem with $Q$ being the grid, and its connections to the Boolean case and to the forbidden submatrix problem.

A Berge-path of length $k$ in a hypergraph $\mathcal{H}$ is a sequence $v_1,e_1,v_2,e_2,\dots,v_{k},e_k,v_{k+1}$ of distinct vertices and hyperedges with $v_{i},v_{i+1} \in e_i$, for $i \le k$. F\"uredi, Kostochka and Luo, and independently Gy\H{o}ri, Salia and Zamora determined the maximum number of hyperedges in an $n$-vertex, connected, $r$-uniform hypergraph that does not contain a Berge-path of length $k$ provided $k$ is large enough compared to $r$. They also determined the unique extremal hypergraph $\mathcal{H}_1$. We prove a stability version of this result by presenting another construction $\mathcal{H}_2$ and showing that any $n$-vertex, connected, $r$-uniform hypergraph without a Berge-path of length $k$, that contains more than $|\mathcal{H}_2|$ hyperedges must be a sub-hypergraph of the extremal hypergraph $\mathcal{H}_1$, provided $k$ is large enough compared to $r$.

The aim of our paper is to generalize the tangent prolongation of Lie groups to non-associative multiplications and to examine how the weak associative and weak inverse properties are transferred to the multiplication defined on the tangent bundle. We obtain that the tangent prolongation of a $\mathcal{C}^r$-differentiable loop ($r\geq 1$) is a $\mathcal{C}^{r-1}$-differentiable loop that acquires the classical weak inverse and weak associative properties of the initial loop.

We study covering numbers and local covering numbers with respect to difference graphs and complete bipartite graphs. In particular we show that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles, and prove that this is best-possible. This answers two questions by Kim, Martin, Masa{\v{r}}{\'\i}k, Shull, Smith, Uzzell, and Wang (Europ. J. Comb. 2020), namely: - What is the local complete bipartite cover number of a difference graph? - Is there a sequence of graphs with constant local difference graph cover number and unbounded local complete bipartite cover number? We add to the study of these local covering numbers with a lower bound construction and some examples. Following Kim \emph{et al.}, we use the results on local covering numbers to provide lower and upper bounds for the local dimension of partially ordered sets of height~2. We discuss the local dimension of some posets related to Boolean lattices and show that the poset induced by the first two layers of the Boolean lattice has local dimension $(1 + o(1))\log_2\log_2 n$. We conclude with some remarks on covering numbers for digraphs and Ferrers dimension., Comment: Parts of this paper have previously been reported in arXiv submission arXiv:1902.08223

In this paper we initiate a systematic study of the Tur\'an problem for edge-ordered graphs. A simple graph is called $\textit{edge-ordered}$, if its edges are linearly ordered. An isomorphism between edge-ordered graphs must respect the edge-order. A subgraph of an edge-ordered graph is itself an edge-ordered graph with the induced edge-order. We say that an edge-ordered graph $G$ $\textit{avoids}$ another edge-ordered graph $H$, if no subgraph of $G$ is isomorphic to $H$. The $\textit{Tur\'an number}$ of an edge-ordered graph $H$ is the maximum number of edges in an edge-ordered graph on $n$ vertices that avoids $H$. We study this problem in general, and establish an Erd\H{o}s-Stone-Simonovits-type theorem for edge-ordered graphs -- we discover that the relevant parameter for the Tur\'an number of an edge-ordered graph is its $\textit{order chromatic number}$. We establish several important properties of this parameter. We also study Tur\'an numbers of edge-ordered paths, star forests and the cycle of length four. We make strong connections to Davenport-Schinzel theory, the theory of forbidden submatrices, and show an application in Discrete Geometry., Comment: 41 pages. Updated grants

Our paper deals with the investigation of extensions of commutative groups by loops so that the quasigroups that result in the multiplication between cosets of the kernel subgroup are T-quasigroups. We limit our study to extensions in which the quasigroups determining the multiplication are linear functions without constant term, called linear abelian extensions. We characterize constructively such extensions with left-, right-, or inverse properties using a general construction according to an equivariant group action principle. We show that the obtained constructions can be simplified for ordered loops. Finally, we apply our characterization to determine the possible cardinalities of the component loop of finite linear abelian extensions.

We are given a set $A$ of buyers, a set $B$ of houses, and for each buyer a preference list, i.e., an ordering of the houses. A house allocation is an injective mapping $\tau$ from $A$ to $B$, and $\tau$ is strictly better than another house allocation $\tau'\neq \tau$ if for every buyer $i$, $\tau'(i)$ does not come before $\tau(i)$ in the preference list of $i$. A house allocation is Pareto optimal if there is no strictly better house allocation. Let $s(\tau)$ be the image of $\tau$ (i.e., the set of houses sold in the house allocation $\tau$). We are interested in the largest possible cardinality $f(m)$ of the family of sets $s(\tau)$ for Pareto optimal mappings $\tau$ taken over all sets of preference lists of $m$ buyers. We improve the earlier upper bound on $f(m)$ given by Asinowski, Keszegh and Miltzow by making a connection between this problem and some problems in extremal set theory.

Suppose that the vertices of a graph $G$ are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists) if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by $m(G)$. It was shown by Saks and Werman that $m(K_n)=n-b(n)$, where $b(n)$ is the number of 1's in the binary representation of $n$. In this paper, we initiate the study of the problem for general graphs. The obvious bounds for a connected graph $G$ on $n$ vertices are $n-b(n)\le m(G)\le n-1$. We show that for any tree $T$ on an even number of vertices we have $m(T)=n-1$ and that for any tree $T$ on an odd number of vertices, we have $n-65\le m(T)\le n-2$. Our proof uses results about the weighted version of the problem for $K_n$, which may be of independent interest. We also exhibit a sequence $G_n$ of graphs with $m(G_n)=n-b(n)$ such that $G_n$ has $O(nb(n))$ edges and $n$ vertices., Comment: 19 pages

In many proofs concerning extremal parameters of Berge hypergraphs one starts with analyzing that part of that shadow graph which is contained in many hyperedges. Capturing this phenomenon we introduce two new types of hypergraphs. A hypergraph $\mathcal{H}$ is a $t$-heavy copy of a graph $F$ if there is a copy of $F$ on its vertex set such that each edge of $F$ is contained in at least $t$ hyperedges of $\mathcal{H}$. $\mathcal{H}$ is a $t$-wise Berge copy of $F$ if additionally for distinct edges of $F$ those $t$ hyperedges are distinct. We extend known upper bounds on the Tur\'an number of Berge hypergraphs to the $t$-wise Berge hypergraphs case. We asymptotically determine the Tur\'an number of $t$-heavy and $t$-wise Berge copies of long paths and cycles and exactly determine the Tur\'an number of $t$-heavy and $t$-wise Berge copies of cliques. In the case of 3-uniform hypergraphs, we consider the problem in more details and obtain additional results., Comment: 20 pages

A widely investigated subject in combinatorial geometry, originated from Erd\H{o}s, is the following. Given a point set $P$ of cardinality $n$ in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. Consider planar arrangements of $n$ lines. Determine the maximum number of triangles of unit area, maximum area or minimum area, determined by these lines. Determine the minimum size of a subset of these $n$ lines so that all triples determine distinct area triangles. We prove that the order of magnitude for the maximum occurrence of unit areas lies between $\Omega(n^2)$ and $O(n^{9/4})$. This result is strongly connected to both additive combinatorial results and Szemer\'edi--Trotter type incidence theorems. Next we show a tight bound for the maximum number of minimum area triangles. Finally we present lower and upper bounds for the maximum area and distinct area problems by combining algebraic, geometric and combinatorial techniques., Comment: Title is shortened. Some typos and small errors were corrected

We study the maximum number $ex(n,e,H)$ of copies of a graph $H$ in graphs with given number of vertices and edges. We show that for any fixed graph $H$, $ex(n,e,H)$ is asymptotically realized by the quasi-clique provided that the edge density is sufficiently large. We also investigate a variant of this problem, when the host graph is bipartite.

In this short note we consider the oriented vertex Tur\'an problem in the hypercube: for a fixed oriented graph $\overrightarrow{F}$, determine the maximum size $ex_v(\overrightarrow{F}, \overrightarrow{Q_n})$ of a subset $U$ of the vertices of the oriented hypercube $\overrightarrow{Q_n}$ such that the induced subgraph $\overrightarrow{Q_n}[U]$ does not contain any copy of $\overrightarrow{F}$. We obtain the exact value of $ex_v(\overrightarrow{P_k}, \overrightarrow{Q_n})$ for the directed path $\overrightarrow{P_k}$, the exact value of $ex_v(\overrightarrow{V_2}, \overrightarrow{Q_n})$ for the directed cherry $\overrightarrow{V_2}$ and the asymptotic value of $ex_v(\overrightarrow{T}, \overrightarrow{Q_n})$ for any directed tree $\overrightarrow{T}$.

A subfamily $\{F_1,F_2,\dots,F_{|P|}\}\subseteq \mathcal F$ is a copy of the poset $P$ if there exists a bijection $i:P\rightarrow \{F_1,F_2,\dots,F_{|P|}\}$ such that $p\le_P q$ implies $i(p)\subseteq i(q)$. A family $\mathcal F$ is $P$-free, if it does not contain a copy of $P$. In this paper we establish basic results on the maximum possible number of $k$-chains in a $P$-free family $\mathcal F\subseteq 2^{[n]}$. We prove that if the height of $P$, $h(P) > k$, then this number is of the order $\Theta(\prod_{i=1}^{k+1}\binom{l_{i-1}}{l_i})$, where $l_0=n$ and $l_1\ge l_2\ge \dots \ge l_{k+1}$ are such that $n-l_1,l_1-l_2,\dots, l_k-l_{k+1},l_{k+1}$ differ by at most one. On the other hand if $h(P)\le k$, then we show that this number is of smaller order of magnitude. Let $\vee_r$ denote the poset on $r+1$ elements $a, b_1, b_2, \ldots, b_r$, where $a < b_i$ for all $1 \le i \le r$ and let $\wedge_r$ denote its dual. For any values of $k$ and $l$, we construct a $\{\wedge_k,\vee_l\}$-free family and we conjecture that it contains asymptotically the maximum number of pairs in containment. We prove that this conjecture holds under the additional assumption that a chain of length 4 is forbidden. Moreover, we prove the conjecture for some small values of $k$ and $l$. We also derive the asymptotics of the maximum number of copies of certain tree posets $T$ of height 2 in $\{\wedge_k,\vee_l\}$-free families $\mathcal F \subseteq 2^{[n]}$.

The maximum size, $La(n,P)$, of a family of subsets of $[n]=\{1,2,...,n\}$ without containing a copy of $P$ as a subposet, has been intensively studied. Let $P$ be a graded poset. We say that a family $\mathcal{F}$ of subsets of $[n]=\{1,2,...,n\}$ contains a \emph{rank-preserving} copy of $P$ if it contains a copy of $P$ such that elements of $P$ having the same rank are mapped to sets of same size in $\mathcal{F}$. The largest size of a family of subsets of $[n]=\{1,2,...,n\}$ without containing a rank-preserving copy of $P$ as a subposet is denoted by $La_{rp}(n,P)$. Clearly, $La(n,P) \le La_{rp}(n,P)$ holds. In this paper we prove asymptotically optimal upper bounds on $La_{rp}(n,P)$ for tree posets of height $2$ and monotone tree posets of height $3$, strengthening a result of Bukh in these cases. We also obtain the exact value of $La_{rp}(n,\{Y_{h,s},Y_{h,s}'\})$ and $La(n,\{Y_{h,s},Y_{h,s}'\})$, where $Y_{h,s}$ denotes the poset on $h+s$ elements $x_1,\dots,x_h,y_1,\dots,y_s$ with $x_1<\dots

Upper bounds to the size of a family of subsets of an n-element set that avoids certain configurations are proved. These forbidden configurations can be described by inclusion patterns and some sets having the same size. Our results are closely related to the forbidden subposet problems, where the avoided configurations are described solely by inclusions., Comment: 19 pages

We investigate the number of 4-edge paths in graphs with a fixed number of vertices and edges. An asymptotically sharp upper bound is given to this quantity. The extremal construction is the quasi-star or the quasi-clique graph, depending on the edge density. An easy lower bound is also proved. This answer resembles the classic theorem of Ahlswede and Katona about the maximal number of 2-edge paths, and a recent theorem of Kenyon, Radin, Ren and Sadun about k-edge stars., Comment: 15 pages, 7 figures

The objectives of this paper is to give a systematic investigation of extension theory of loops. A loop extension is (left, right or middle) nuclear, if the kernel of the extension consists of elements associating (from left, right or middle) with all elements of the loop. It turns out that the natural non-associative generalizations of the Schreier's theory of group extensions can be characterized by different types of nuclear properties. Our loop constructions are illustrated by rich families of examples in important loop classes., Comment: 21 pages

We study the relationship between the sizes of two sets $B, S\subset\mathbb{R}^2$ when $B$ contains either the whole boundary, or the four vertices, of a square with axes-parallel sides and center in every point of $S$, where size refers to one of cardinality, Hausdorff dimension, packing dimension, or upper or lower box dimension. Perhaps surprinsingly, the results vary depending on the notion of size under consideration. For example, we construct a compact set $B$ of Hausdorff dimension $1$ which contains the boundary of an axes-parallel square with center in every point $[0,1]^2$, but prove that such a $B$ must have packing and lower box dimension at least $\tfrac{7}{4}$, and show by example that this is sharp. For more general sets of centers, the answers for packing and box counting dimensions also differ. These problems are inspired by the analogous problems for circles that were investigated by Bourgain, Marstrand and Wolff, among others., Comment: 20 pages, no figures

Three intersection theorems are proved. First, we determine the size of the largest set system, where the system of the pairwise unions is l-intersecting. Then we investigate set systems where the union of any s sets intersect the union of any t sets. The maximal size of such a set system is determined exactly if s+t<5, and asymptotically if s+t>4. Finally, we exactly determine the maximal size of a k-uniform set system that has the above described (s,t)-union-intersecting property, for large enough n., Comment: 9 pages

Let $B_n$ be the poset generated by the subsets of $[n]$ with the inclusion as relation and let $P$ be a finite poset. We want to embed $P$ into $B_n$ as many times as possible such that the subsets in different copies are incomparable. The maximum number of such embeddings is asymptotically determined for all finite posets $P$ as $\frac{{n \choose \lfloor n/2\rfloor}}{M(P)}$, where $M(P)$ denotes the minimal size of the convex hull of a copy of $P$. We discuss both weak and strong (induced) embeddings., Comment: 8 pages

In this paper we prove that the holonomy group of a simply connected locally projectively flat Finsler manifold of constant curvature is a finite dimensional Lie group if and only if it is flat or it is Riemannian., Comment: 9 pages

Our goal in this paper is to make an attempt to find the largest Lie algebra of vector fields on the indicatrix such that all its elements are tangent to the holonomy group of a Finsler manifold. First, we introduce the notion of the curvature algebra, generated by curvature vector fields, then we define the infinitesimal holonomy algebra by the smallest Lie algebra of vector fields on an indicatrix, containing the curvature vector fields and their horizontal covariant derivatives with respect to the Berwald connection. At the end we introduce the notion of the holonomy algebra of a Finsler manifold by all conjugates of infinitesimal holonomy algebras by parallel translations with respect to the Berwald connection. We prove that this holonomy algebra is tangent to the holonomy group.

In this paper we show that the topological closure of the holonomy group of a certain class of projectively flat Finsler 2-manifolds of constant curvature is maximal, that is isomorphic to the connected component of the diffeomorphism group of the circle. This class of 2-manifolds contains the standard Funk plane of constant negative curvature and the Bryant-Shen-spheres of constant positive curvature. The result provides the first examples describing completely infinite dimensional Finslerian holonomy structures.

For a given finite poset $P$, $La(n,P)$ denotes the largest size of a family $\mathcal{F}$ of subsets of $[n]$ not containing $P$ as a weak subposet. We exactly determine $La(n,P)$ for infinitely many $P$ posets. These posets are built from seven base posets using two operations. For arbitrary posets, an upper bound is given for $La(n,P)$ depending on $|P|$ and the size of the longest chain in $P$. To prove these theorems we introduce a new method, counting the intersections of $\mathcal{F}$ with double chains, rather than chains., Comment: 8 pages, 5 figures. Submitted to The Electronic Journal of Graph Theory and Applications (EJGTA)

Recently, we developed a method for the study of holonomy properties of non-Riemannian Finsler manifolds and obtained that the holonomy group can not be a compact Lie group, if the Finsler manifold of dimension $> 2$ has non-zero constant flag curvature. The purpose of this paper is to move further, exploring the holonomy properties of projectively flat Finsler manifolds of non-zero constant flag curvature. We prove in particular that projectively flat Randers and Bryant-Shen manifolds of non-zero constant flag curvature have infinite dimensional holonomy group., Comment: 15 pages

Our paper is devoted to the study of the holonomy groups of Finsler surfaces using the methods of infinite dimensional Lie theory. The notion of infinitesimal holonomy algebra will be introduced, by the smallest Lie algebra of vector fields on an indicatrix, containing the curvature vector fields and their horizontal covariant derivatives with respect to the Berwald connection. We obtain that the topological closure of the holonomy group contains the exponential image of any tangent Lie algebra of the holonomy group. A class of Randers surfaces is determined, for which the infinitesimal holonomy algebra coincides with the curvature algebra. We prove that for all projectively flat Randers surfaces of non-zero constant flag curvature the infinitesimal holonomy algebra has infinite dimension and hence the holonomy group cannot be a Lie group of finite dimension. Finally, in the case of the Funk metric we prove that the infinitesimal holonomy algebra is a dense subalgebra of the Lie algebra of the full diffeomorphism group and hence the topological closure of the holonomy group is the orientation preserving diffeomorphism group of the circle., Comment: 16 pages

The aim of this paper is to show that holonomy properties of Finsler manifolds can be very different from those of Riemannian manifolds. We prove that the holonomy group of a positive definite non-Riemannian Finsler manifold of non-zero constant curvature with dimension >2 cannot be a compact Lie group. Hence this holonomy group does not occur as the holonomy group of any Riemannian manifold. In addition, we provide an example of left invariant Finsler metric on the Heisenberg group, so that its holonomy group is not a (finite dimensional) Lie group. These results give a positive answer to the following problem formulated by S. S. Chern and Z. Shen: "Is there a Finsler manifold whose holonomy group is not the holonomy group of any Riemannian manifold?", Comment: to appear in Houston Journal of Mathematics

In the year 1944 R. H. Bruck has described a very general construction method which he called the extension of a set by a quasigroup. We use it to construct a class of examples for LF-quasigroups in which the image of the map $e(x)=x\backslash x$ is a group. More generally, we consider the variety of quasigroups which is defined by the property that the map $e$ is an endomorphism and its subvariety where the image of the map $e$ is a group. We characterize quasigroups belonging to these varieties using their Bruck decomposition with respect to the map $e$., Comment: to appear in Journal of Generalized Lie Theory and Applications