Disjoint edges in topological graphs and the tangled-thrackle conjecture

It is shown that for a constant $t\in \mathbb{N}$, every simple topological graph on $n$ vertices has $O(n)$ edges if it has no two sets of $t$ edges such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is $K_{t,t}$-free). As an application, we settle the \emph{tangled-thrackle} conjecture formulated by Pach, Radoi\v{c}i\'c, and T\'oth: Every $n$-vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most $O(n)$ edges

Significance of the Climate and Energy Package for the Development of Renewable Energy Sources in the European Union

The Climate and Energy Package obliges member states to pursue a common aim - to increase the share of renewable energy to 20% in the general energy balance of the European Union by 2020. This is a challenging task, since the renewable energy sector requires significant financial support to increase its competitiveness, compared to traditional energy sources. When adopting the Energy Package, leaders of member states did not anticipate the economic crisis and its impact on the European Union’s economy, and on the energy sector in particular.Pakiet energetyczno- klimatyczny zobligował kraje członkowskie do spełnienia wspólnego celu jakim jest zwiększenie udziału energii z odnawialnych źródeł energii do 20% w ogólnym bilansie energetycznym UE do 2020.r Jest to zadanie niezwykle trudne gdyż sektor odnawialnych źródeł wymaga rozwoju i ogromnego wsparcia finansowego, w celu zwiększenia jego konkurencyjności w stosunku do źródeł konwencjonalnych energii. Przyjmując pakiet energetyczny przywódcy państw członkowskich nie przewidzieli kryzysu gospodarczego i jego wpływu na gospodarkę Unii Europejskiej, w tym na sektor energetyczny

Poland’s Energy Security in Light of a Statistical Analysis

In the research on Poland’s energy security compared to the European Union, data clustering was carried out according to the following methods: the Ward method, the full bond (furthest neighbourhood) method, analysis using the k-means method, and multidimensional scaling. In order to define the shifts in the direction of Poland’s energy security, synthetic indices for 2000 and 2008 have been calculated. Obtaining the answers to the above questions allowed for defining the main trends in Poland’s activities aimed at increasing its energy security.W badaniach nad bezpieczeństwem energetycznym Polski na tle Unii Europejskiej została zastosowana analiza skupień przeprowadzona metodą Warda i metodą pełnego wiązania (najdalszego sąsiedztwa) oraz analiza metodą k-średnich i skalowanie wielowymiarowe. W celu uchwycenia kierunków zmian bezpieczeństwa energetycznego Polski zostały obliczone wskaźniki syntetyczne dla 2000 r. i 2008 r. Uzyskanie odpowiedzi na powyższe pytania pozwoliło określić główne kierunki działania Polski na rzecz zwiększenia bezpieczeństwa energetycznego kraju

Applications of a new separator theorem for string graphs

An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphs G with n vertices: (i) if K_t is not a subgraph of G for some t, then the chromatic number of G is at most (\log n)^{O(\log t)}; (ii) if K_{t,t} is not a subgraph of G, then G has at most t(\log t)^{O(1)}n edges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdos-Hajnal conjecture almost holds for string graphs.Comment: 7 page

Cross-intersecting families of vectors

Given a sequence of positive integers $p = (p_1, . . ., p_n)$, let $S_p$ denote the family of all sequences of positive integers $x = (x_1,...,x_n)$ such that $x_i \le p_i$ for all $i$. Two families of sequences (or vectors), $A,B \subseteq S_p$, are said to be $r$-cross-intersecting if no matter how we select $x \in A$ and $y \in B$, there are at least $r$ distinct indices $i$ such that $x_i = y_i$. We determine the maximum value of $|A|\cdot|B|$ over all pairs of $r$- cross-intersecting families and characterize the extremal pairs for $r \ge 1$, provided that $\min p_i >r+1$. The case $\min p_i \le r+1$ is quite different. For this case, we have a conjecture, which we can verify under additional assumptions. Our results generalize and strengthen several previous results by Berge, Frankl, F\"uredi, Livingston, Moon, and Tokushige, and answers a question of Zhang

Decomposition of multiple packings with subquadratic union complexity

Suppose $k$ is a positive integer and $\mathcal{X}$ is a $k$-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most $k$ sets. Suppose there is a function $f(n)=o(n^2)$ with the property that any $n$ members of $\mathcal{X}$ determine at most $f(n)$ holes, which means that the complement of their union has at most $f(n)$ bounded connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that $\mathcal{X}$ can be decomposed into at most $p$ ($1$-fold) packings, where $p$ is a constant depending only on $k$ and $f$.Comment: Small generalization of the main result, improvements in the proofs, minor correction