The length polyhedron QP of an interval order P is the convex hull of integral vectors representing the interval lengths in interval representations of P. This polyhedron has been studied by various authors, including Fishburn and Isaak. Notably, QP forms a pointed affine cone, a property inherited from being a projection of the representation polyhedron, a structure explored also by Doignon and Pauwels. The apex of the length polyhedron corresponds to Greenough’s minimal endpoint representation, which is, in fact, the length vector of the canonical interval representation—an interval representation that minimizes the sum of the interval lengths.Building on a combinatorial perspective of canonical representations, we refine Isaak’s graph-theoretical model by introducing a new and simpler directed graph. From directed cycles of this key digraph, we extract a linear system of inequalities that precisely characterizes the length polyhedron QP. This combinatorial approach also reveals the unique Hilbert basis of the polyhedron, which consists of binary rays. We prove that the intersection graph of the sets corresponding to these binary rays are Berge graphs; therefore they are perfect graphs. As a result, for interval orders with bounded width, the length polyhedron has a polynomial-sized Hilbert basis, which can be computed in polynomial time. We also provide an example of a family of interval orders with a Hilbert basis of exponential size. In a companion paper we determine the Schrijver system for the length polyhedron. We conclude with open problems.

We study the metric dimension (strong and weak) of infinite graphs. In particular, our main interest is characterizing infinite graphs with finite dimension. Our main results: (1) graphs with more than one end have infinite strong dimension; (2) for graphs with a finite number of cycles, the weak dimension is finite if and only if the graph has finitely many vertices of degree three, and the strong dimension is finite if and only if the graph has one end and finitely many vertices of degree three.

In this paper, we present a graph-based analysis of the topology of D-Wave quantum computers, focusing on the Pegasus, Chimera, and Zephyr architectures. We investigate these topologies under different parameter settings using k-hop-based graph metrics. Each of these architectures comprises distinct subgraphs in which qubits are interconnected according to specific patterns dictated by their implementation. Our study pursues two primary objectives. First, we analyze the structural properties of the Chimera, Pegasus, and Zephyr topologies, examining their scalability and connectivity characteristics. Second, we evaluate the behavior of graph-based density and redundancy metrics within these architectures. The inherent symmetries of these quantum hardware designs provide a unique opportunity to systematically assess the effectiveness of these metrics across varying connectivity patterns. By leveraging these symmetries, our findings not only enhance the understanding of these topological structures but also offer deeper insights into the reliability and applicability of the proposed metrics in the broader context of quantum hardware design.

The interval count problem, a classical question in the study of interval orders, was introduced by Ronald Graham in the 1980s. This problem asks: given an interval order P, what is the minimum number of distinct interval lengths required to construct an interval representation of P? Interval orders that can be represented with just one interval length are known as semiorders, and their characterization is well known. However, the characterization of interval orders that require at most k interval lengths — termed k-count interval orders—remains an open and challenging problem for k≥ 2 . Our investigation into 2-count interval orders led us naturally to consider a related problem, interval representations of permutations, which we introduce in this paper. Specifically, we characterize permutations that have a 2-count interval representation. We prove that a permutation admits a 2-count interval representation if and only if its longest decreasing subsequences have length at most 2. For larger values of k, however, a similar characterization does not hold. There are permutations that do not permit a 3-count interval representation even though all of their decreasing subsequences have length at most 3. Characterizing k-count permutations remains open for k ≥ 3 . The k-count permutation representation problem appears to capture essential aspects of the broader problem of characterizing k-count interval orders. To support this connection, we apply our findings on interval representations of permutations to demonstrate that a height-3 interval order is 2-count if and only if it has depth at most 2, where the depth of an interval order is the minimum, over all interval representations of the order, of the maximum number of intervals in a chain of nested intervals.

Graph analysis has long relied on classical metrics such as degree distribution, centrality measures, and clustering coefficients. However, these traditional metrics often fail to capture the structural variability in local and global connectivity patterns. In this paper, we introduce the k-hop entropy metric (KHEM), a structure-sensitive measure that quantifies the complexity of a node's k-hop neighborhood. Unlike degree-based measures, KHEM incorporates the entropy of node degree distributions within a given neighborhood radius, providing deeper insight into network heterogeneity. An analysis of KHEM has been conducted across various network models, including Erdos-Renyi random graphs, Barabasi-Albert scale-free networks, Watts-Strogatz small-world graphs, and quantum computing topologies such as D-Wave's architectures. The findings indicate that KHEM provides a more nuanced representation of local structural complexity by distinguishing nodes according to neighborhood entropy rather than mere connectivity. This leads to enhanced node ranking and a more refined structural characterization across diverse types of networks.

Two hybrid, entropy-guided node metrics are proposed for the k-hop environment: Entropy-Weighted Redundancy (EWR) and Normalized Entropy Density (NED). The central idea is to couple local Shannon entropy with neighborhood density/redundancy so that structural heterogeneity around a vertex is captured even when classical indices (e.g., degree or clustering) are similar. The metrics are formally defined and shown to be bounded, isomorphism-invariant, and stable under small edge edits. Their behavior is assessed on representative topologies (Erdős–Rényi, Barabási–Albert, Watts–Strogatz, random geometric graphs, and the Zephyr quantum architecture). Across these settings, EWR and NED display predominantly negative correlation with degree and provide information largely orthogonal to standard centralities; vertices with identical degree can differ by factors of two to three in the proposed scores, revealing bridges and heterogeneous regions. These properties indicate utility for vulnerability assessment, topology-aware optimization, and layout heuristics in engineered and quantum networks.

We investigate how the metric dimension of infinite graphs change when we add edges to the graph. Our two main results: (1) there exists a growing sequence of graphs (under the subgraph relation, but without adding vertices) for which the metric dimension changes between finite and infinite infinitely many times; (2) finite changes in the edge set can not change the metric dimension from finite to infinite or vice versa.

Representations of interval orders are, in general, may use an arbitrary set of interval lengths. We can define subclasses of interval orders by restricting the allowable lengths of intervals. Motivated by a recent paper of Keller, Trenk, and Young, we study the dimension of posets in some of these subclasses. Among other results, we answer several of their questions, and we simplify the proof of one of their main results.

In this paper, we focus on graph -based analysis of the topology of D -Wave quantum computers. The Pegasus, Chimera and Zephyr topologies generated with different parameters are examined using classical based graph metrics. Our main goal is to use metrics to highlight the main features and limitations of these topologies. The secondary goal is that the results contribute to the further development of more efficient quantum processors.

In this paper, we focus on graph-based analysis of the topology of $D$-Wave quantum computers. The Pegasus, Chimera, and Zephyr topologies generated with different parameters are examined using $k$-hop-based graph metrics. In addition to the well-known classical graph metrics, we briefly describe the density and redundancy-based metrics interpreted in the $k$-hop environment. These topologies consist of different subgraphs depending on the implementation in which the qubits are connected in a specific pattern. Our main goal is to use metrics to highlight the main features and limitations of these topologies. The secondary goal is that the results contribute to the further development of more efficient quantum processors.

A magyar törvénykezésben megjelentek azok a rendeletek, amelyek ráerősítenek arra, hogy gyorsan közeledünk a kvantumszámítógépek korszakához. A teljesen újszerű elveken működő kvantumszámítógépek korábban nem megoldható problémákra hatékony és gyors megoldásokat ígérnek. Az új típusú gépek megjelenése várhatóan nem csak az informatika, a gazdaság és a különféle tudományok területén hoz nagy változásokat, hanem valószínűsíthetően a mindennapi életünkre is erősen hatással lesz. Szükségét érezzük, hogy az oktatási szféra is kövesse a kvantumtechnológia megjelenését, hogy ne érje váratlanul a jövő szakembereit. A tanulmányban szóba kerülnek a témával kapcsolatos jogszabályok, az érintett szakterületek és a kvantuminformatikában rejlő lehetőségek és veszélyek. Továbbá írunk arról, hogy milyen tapasztalatok vannak eddig a kvantuminformatika tanításának gyakorlatában.

An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemerédi proved that any on-line algorithm could be forced to use ( [ w+1; 2 ]) chains to partition a poset of width w. The maximum number of chains that can be forced on any on-line algorithm remains unknown. In a survey paper by Bosek et al., it is shown that Szemerédi’s argument could be improved to obtain a lower bound almost twice as good. Variants of the problem were considered where the class is restricted to posets of bounded dimension or where the poset is presented via a realizer of size d. In this paper, we prove two results. First, we prove that any on-line algorithm can be forced to use (2-o(1))( [ w+1; 2 ]) chains to partition a 2-dimensional poset of width w. Second, we prove that any on-line algorithm can be forced to use (2-1/d-1-o(1))( [ w+1; 2 ]) chains to partition a poset of width w presented via a realizer of size d.

The development of quantum computers is bringing major changes to the IT sector. These computers, based on completely new principles, can provide effective solutions to previously unsolvable problems. Regulations have already been introduced in the European Union and Hungarian law to confirm that we are getting closer to the era of quantum computers. Therefore, we believe that education and teachers should follow the development of these machines so that future students in the field of information technology, whether they are IT teachers, physicists, or programmers, are not caught unawares. In this article, we present some examples from abroad where quantum computing topics are already included at certain levels of education.

Let H be a complete r -uniform hypergraph such that two vertices are marked in each edge as its ‘boundary’ vertices. A linear ordering of the vertex set of H is called an agreeing linear order , provided all vertices of each edge of H lie between its two boundary vertices. We prove the following Helly-type theorem: if there is an agreeing linear order on the vertex set of every subhypergraph of H with at most 2 r − 2 vertices, then there is an agreeing linear order on the vertex set of H . We also show that the constant 2 r − 2 cannot be reduced in the theorem. The case r = 3 of the theorem has particular interest in the axiomatic theory of betweenness. Similar results are obtained for further r -uniform hypergraphs ( r ≥ 3), where one or two vertices are marked in each edge, and the linear orders need to satisfy various rules of agreement. In one of the cases we prove that no such Helly-type statement holds.

A Szerzők egy 25 éves nőbeteg poszttraumás felső végtagra és nyaki régióra kiterjedő recidív hajlamú, gravis subcutan emphysemával járó megbetegedését mutatják be. Felvétele napján banális traumát követően bal felső végtagon subcutan emphysema jelentkezett egyéb szisztémás eltérés nélkül. A beteg rapidan romló általános állapota miatt műtét vált szükségessé. Az emphysema kiterjedése fokozatosan nőtt, mielőtt stagnálni, végül regrediálni kezdett volna. Gyógyultan emittálták, az egyhetes kontrollvizsgálaton recidíva, panasz nem került említésre. A páciens emissziót követően egy hónap múlva ismét jelentkezett, ahol az előző alkalomhoz képest proximalisan elhelyezkedő, a bal felső végtag egészére és a nyaki régióra is kiterjedő subcutan emphysema igazolódott. A feltárás, necrectomia ellenére a beteg általános állapota romlani kezdett. Laborértékek szignifikáns eltérést nem mutattak, gázképző kórokozót nem lehetett kimutatni. Kétnaponta sebrevíziók történtek, az operatív és intravénás antibiotikus terápiát nagy áramlású maszkos oxigénterápiával egészítették ki. A beteg általános állapota végül javulni kezdett, sebei reakciómentesen gyógyultak, a subcutan emphysema felszívódott, nem recidivált. Gyógyultan emittálták, kontrollvizsgálatokon megjelent, panasza azóta nincsen.

An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains in the partition. The on-line chain partitioning problem involves finding the minimal number of chains needed by an on-line algorithm. Chrobak and Ślusarek considered variants of the on-line chain partitioning problem in which the elements are presented as intervals and intersecting intervals are incomparable. They constructed an on-line algorithm which uses at most 3w−2 chains, where w is the width of the interval order, and showed that this algorithm is optimal. They also considered the problem restricted to intervals of unit-length and while they showed that first-fit needs at most 2w−1 chains, over 30 years later, it remains unknown whether this is an optimal algorithm. In this paper, we improve upon previously known bounds and show that any on-line algorithm can be forced to use ⌈32w⌉ chains to partition an order presented in the form of its unit interval representation. As a consequence, we completely solve the problem for w=3. Lastly, we show that loosening the restriction from unit intervals to proper intervals in the bandwidth variant allows us to improve the lower bound by w/3.

The converse of the Borel-Cantelli Lemma states that if {A(i) }(i-1)(infinity)events such that n-ary sumation sigma(A(i)) = infinity, then almost surely infinitely many of these events will occur. i=1 is a sequence of independent Erdos and Renyi proved that it is sufficient to weaken the condition of independence to pairwise independence. Later, several other weakenings of the condition appeared in the literature. The aim of this paper is to provide a collection of conditions, all of which imply that almost surely infinitely many of the events occur, and determine the complete implicational relationship between them. Many of these results are known, or follow from known results, however, they are not widely known among non-specialists. Yet, the results can be extremely useful for areas outside of probability theory, as evidenced by the original motivation of this paper emerging from infinite combinatorics. Our proofs are aimed to be accessible to a general mathematical audience. ?c 2021 Elsevier GmbH. All rights reserved.

Összefoglalás. A biztonságra evidenciaként tekint az utazó a desztinációválasztás során. Annak tartalma erősen szubjektív, egyénenként eltérő szintet képvisel. A COVID–19-járvány időszakában megvalósult kutatásunk válasszal kívánt szolgálni többek között arra, hogy az időskorú német utazók esetében mely faktorok határozzák meg leginkább a desztináció preferenciát, és ez hogyan tükröződik a költésükben. 2021. július–október között megvalósításra került személyes megkérdezés végső mintáját 347 fő (55+ éves) német szenior utazó adta. Az eredmények rámutattak, hogy a marketingtudomány által, általánosan homogénként kezelt fogyasztói szegmens további alszegmensekre bontható, tipizálható, továbbá a desztináció preferenciát esetükben leginkább az infrastruktúra minősége és a biztonság határozza meg. Summary. Introduction: Tourism does not exist without security. In Maslow’s hierarchy of needs, security is the second most determinant aspect after physiological concerns. We consider that as an evident fact, thus the secure nature of a destination is an obvious expectation regarding travelling aspects. The measure and actual meaning of this latter component is quite subjective, thus it varies person by person. Security can be investigated from several aspects, so it is important to highlight that our sample was analysed in relation to health security concerns. Investigation materials and methods: In our research, conducted during the period of the COVID-19 pandemic, we pursued to outline the role of security, as well as the so-called Corona-Protocol, which was established for prevention purposes regarding the chosen destinations in the case of elderly travellers, and to find out how it is reflected in their touristic spending. We conducted personal interviews (PAPI) between July and October 2021 with the participation of 419 people, whose final sample – after being filtered regarding age and nationality aspects – consisted of 347 German senior (age group: 55+) travellers. The survey was carried out on three locations within Hungary (Pécs, Kalocsa, Budapest) with the participation of coordinators. Research outcomes: The received replies were analysed – at first, the 48 touristic variables were submitted to factor analysis, which helped in the allocation of main components. By the use of these main components, we identified further sub-segments (traveller types) within the segments, which gave a superb illustration about the inaccurate nature of the practice, when senior travellers are considered as a homogenous group. In the case of five clusters – despite similar age averages – we identified distinct characteristics concerning the aspects of qualifications, incomes and the measure of touristic expenses, while the priorities regarding destination expectations, security, infrastructure and pricing were also different. Despite their relatively weak explanation capacities, our outcomes – received by regression modelling – pointed out that regarding our sample, the measure of touristic expenses was mostly determined by the infrastructural quality of destination and security. Thereby we can answer the question how much the cost of security for German senior travellers is. Furthermore, another outcome was the realisation that different habitat locations within the same home country have significant impact on the spending of German senior travellers. Our research could constitute an appropriate basis for further research regarding the investigated subject and traveller segment.

Let k denote the totally ordered set (or chain) on k elements. The product k(t) = k x ... x k is a poset called a grid. This paper discusses several loosely related results on the Ramsey theory of grids. Most of the results involve some application of the Product Ramsey Theorem.

INTRODUCTION:The authors audited the outcomes of various surgical interventions employed to fix the dislocations of the acromioclavicular joint in their unit. This resulted in changing their method to treat this condition.OBJECTIVE:Finding a technique that results in the best functional stability of the injured joint while preserving its physiological function.METHOD:In this particular area of traumatology, there is still uncertainty of the ideal way to provide the best outcome in terms of stability and function. From the currently known techniques, the authors have chosen the Minimally INvasive Acromioclavicular Joint Reconstruction (MINAR) method to be introduced in their unit. Its advantage is that it provides good stability combined with fast rehabilitation with a minimally invasive intervention. Furthermore, it is safe and easy to learn and master. It is also to be pointed out that it restores not only the stability but also the physiological functionality of the damaged joint. The authors carried out this type of operation first in 2012. They promoted the technique first in Hungary and became a 'reference centre' in the country. The authors will detail the method itself, both its indications and contraindications as well as the stages of the postoperative rehabilitation.RESULTS:The authors scrutinized their results with imaging and 7 years of retrospective analysis with Constant and DASH scores. These were all in support of their choice of the MINAR method.DISCUSSION:The authors concluded results congruent with international studies and recommendations. Additional analysis deducted aspects of the technique that could lead to further improvement in outcomes and highlighted areas for future study.CONCLUSION:In line with the recommendation of the European Shoulder Associates, the author's recommendation is that the MINAR technique should be the first choice of intervention for dislocations of the acromioclavicular joint. Orv Hetil. 2022; 163(50): 1992-1999.