Vectors in a Box

For an integer d>=1, let tau(d) be the smallest integer with the following property: If v1,v2,...,vt is a sequence of t>=2 vectors in [-1,1]^d with v1+v2+...+vt in [-1,1]^d, then there is a subset S of {1,2,...,t} of indices, 2<=|S|<=tau(d), such that \sum_{i\in S} vi is in [-1,1]^d. The quantity tau(d) was introduced by Dash, Fukasawa, and G\"unl\"uk, who showed that tau(2)=2, tau(3)=4, and tau(d)=Omega(2^d), and asked whether tau(d) is finite for all d. Using the Steinitz lemma, in a quantitative version due to Grinberg and Sevastyanov, we prove an upper bound of tau(d) <= d^{d+o(d)}, and based on a construction of Alon and Vu, whose main idea goes back to Hastad, we obtain a lower bound of tau(d)>= d^{d/2-o(d)}. These results contribute to understanding the master equality polyhedron with multiple rows defined by Dash et al., which is a "universal" polyhedron encoding valid cutting planes for integer programs (this line of research was started by Gomory in the late 1960s). In particular, the upper bound on tau(d) implies a pseudo-polynomial running time for an algorithm of Dash et al. for integer programming with a fixed number of constraints. The algorithm consists in solving a linear program, and it provides an alternative to a 1981 dynamic programming algorithm of Papadimitriou.Comment: 12 pages, 1 figur

Advantage in the discrete Voronoi game

We study the discrete Voronoi game, where two players alternately claim vertices of a graph for t rounds. In the end, the remaining vertices are divided such that each player receives the vertices that are closer to his or her claimed vertices. We prove that there are graphs for which the second player gets almost all vertices in this game, but this is not possible for bounded-degree graphs. For trees, the first player can get at least one quarter of the vertices, and we give examples where she can get only little more than one third of them. We make some general observations, relating the result with many rounds to the result for the one-round game on the same graph

Unsplittable coverings in the plane

A system of sets forms an {\em $m$-fold covering} of a set $X$ if every point of $X$ belongs to at least $m$ of its members. A $1$-fold covering is called a {\em covering}. The problem of splitting multiple coverings into several coverings was motivated by classical density estimates for {\em sphere packings} as well as by the {\em planar sensor cover problem}. It has been the prevailing conjecture for 35 years (settled in many special cases) that for every plane convex body $C$, there exists a constant $m=m(C)$ such that every $m$-fold covering of the plane with translates of $C$ splits into $2$ coverings. In the present paper, it is proved that this conjecture is false for the unit disk. The proof can be generalized to construct, for every $m$, an unsplittable $m$-fold covering of the plane with translates of any open convex body $C$ which has a smooth boundary with everywhere {\em positive curvature}. Somewhat surprisingly, {\em unbounded} open convex sets $C$ do not misbehave, they satisfy the conjecture: every $3$-fold covering of any region of the plane by translates of such a set $C$ splits into two coverings. To establish this result, we prove a general coloring theorem for hypergraphs of a special type: {\em shift-chains}. We also show that there is a constant $c>0$ such that, for any positive integer $m$, every $m$-fold covering of a region with unit disks splits into two coverings, provided that every point is covered by {\em at most} $c2^{m/2}$ sets

An abstract approach to polychromatic coloring

The goal of this paper is to give a new, abstract approach to cover-decomposition and polychromatic colorings using hypergraphs on ordered vertex sets. We introduce an abstract version of a framework by Smorodinsky and Yuditsky, used for polychromatic coloring halfplanes, and apply it to so-called ABA-free hypergraphs, which are a generalization of interval graphs. Using our methods, we prove that (2k−1)-uniform ABA-free hypergraphs have a polychromatic k-coloring, a problem posed by the second author. We also prove the same for hypergraphs defined on a point set by pseudohalfplanes. These results are best possible. We also introduce several new notions that seem to be important for investigating polychromatic colorings and ϵ -nets, such as shallow hitting sets. We pose several open problems related to them. For example, is it true that given a finite point set S on a sphere and a set of halfspheres F, such that {S ∩ F | F ∈ F} is a Sperner family, we can select an R ⊂ S such that 1 ≤ |F ∩ R| ≤ 2 holds for every F ∈ F?. © Springer International Publishing Switzerland 2016

EDGE ORDERED TURAN PROBLEMS

We introduce the Turan problem for edge ordered graphs. We call a simple graph 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 avoids another edge ordered graph H, if no subgraph of G is isomorphic to H. The Turan number ex(<)'(n, H) of a family H of edge ordered graphs is the maximum number of edges in an edge ordered graph on n vertices that avoids all elements of H.We examine this parameter in general and also for several singleton families of edge orders of certain small specific graphs, like star forests, short paths and the cycle of length four

Wpływ różnych poziomów aktywności fizycznej na dyspozycyjną uważność, cechę lęku i cechę agresji

Background. Regular sporting activity can lead to favorable personality changes in addition to positive psychological effects. Our goal was to examine and compare university freshmen with differing sporting habits, so we measured athletes who are competitors (1), regularly active but non-competitor athletes (2) and inactive students (3). Material and methods. We conducted a cross-sectional study among volunteer university freshmen (mean age 18.98 years) from the Faculty of Health Sciences, University of Pécs (Hungary) (n=109). We used self-edited sociodemographic and sporting habits questions and validated, standardized paper-and-pencil tests: Spielberger State Trait Anxiety Inventory, the Mindfulness Attention and Awareness Scale, and the Buss and Perry’s Aggression Questionnaire. Results. Using an independent sample T-test, we found that athletes who are competitors (1) showed significantly higher dispositional mindfulness levels (t=-2.050; p=.043) and significantly lower anxiety levels (t=3.370; p=.001) than the inactive group (3). Considering trait aggression, we found significant difference only in the subscale anger among those students who practice sport regularly and those who are inactive (p=.050, Z=-1.933). The trait aggression total score did not exhibit a relationship with sporting activity in our sample. Conclusions. Intensive and regular physical activity facilitates psychological factors which support individual well-being.Wprowadzenie. Regularne uprawianie sportu może powodować pozytywne zmiany w osobowości oraz mieć korzystny wpływ na psychologię człowieka. Celem analizy podjętej w niniejszej pracy było zbadanie i porównanie studentów pierwszego roku studiów, którzy różnili się od siebie pod kątem nawyków związanych z uprawianiem sportu. Badaniu poddano atletów, którzy biorą udział w zawodach (1), studentów regularnie uprawiających sport, ale niebiorących udziału w zawodach (2) oraz studentów nieaktywnych fizycznie (3). Materiał i metody. Przeprowadzono przekrojowe badanie chętnych studentów pierwszego roku (średni wiek: 18,98 roku) z Wydziału Nauk o Zdrowiu Uniwersytetu w Peczu (Węgry) (n=109). Użyto w nim pytań socjodemograficznych zredagowanych przez autorów badania oraz pytań dotyczących nawyków sportowych, a także zatwierdzonych i standaryzowanych testów w formie papierowej: Inwentarza Stanu i Cechy Lęku (STAI), Skali Świadomej Obecności (MAAS) oraz Kwestionariusza Agresji Bussa i Perry’ego. Wyniki. Po niezależnej próbie testu T zauważono, że atleci biorący udział w zawodach (1) mają znacząco wyższy poziom dyspozycyjnej uważności (t=-2,050; p=0,043) i zdecydowanie niższy poziom lęku (t=3,370; p=0,001) niż grupa studentów nieaktywnych fizycznie (3). Biorąc pod uwagę cechę agresji, odkryto znaczące różnice jedynie w podskali gniewu w przypadku tych studentów, którzy regularnie uprawiają sport, oraz tych, którzy nie są aktywni fizycznie (p=0,050, Z=-1,933). Wynik całkowity cechy agresji nie wykazał jej związków z aktywnością fizyczną w badanej próbie. Wnioski. Intensywna i regularna aktywność fizyczna wspiera rozwój cech psychologicznych, które wzmacniają dobre samopoczucie badanych