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 Radoslav Fulek Computer Science , 2013, Abstract: A topological graph drawn on a cylinder whose base is horizontal is \emph{angularly monotone} if every vertical line intersects every edge at most once. Let $c(n)$ denote the maximum number $c$ such that every simple angularly monotone drawing of a complete graph on $n$ vertices contains at least $c$ pairwise disjoint edges. We show that for every simple complete topological graph $G$ there exists $\Delta$, $0<\Delta  Computer Science , 2014, Abstract: It is shown that for a constant$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)$edges.  Mathematics , 2014, Abstract: A well-known result of Kupitz from 1982 asserts that the maximal number of edges in a convex geometric graph (CGG) on$n$vertices that does not contain$k+1$pairwise disjoint edges is$kn$(provided$n>2k$). For$k=1$and$k=n/2-1$, the extremal examples are completely characterized. For all other values of$k$, the structure of the extremal examples is far from known: their total number is unknown, and only a few classes of examples were presented, that are almost symmetric, consisting roughly of the$kn$"longest possible" edges of$CK(n)$, the complete CGG of order$n$. In order to understand further the structure of the extremal examples, we present a class of extremal examples that lie at the other end of the spectrum. Namely, we break the symmetry by requiring that, in addition, the graph admit an independent set that consists of$q$consecutive vertices on the boundary of the convex hull. We show that such graphs exist as long as$q \leq n-2k$and that this value of$q$is optimal. We generalize our discussion to the following question: what is the maximal possible number$f(n,k,q)$of edges in a CGG on$n$vertices that does not contain$k+1$pairwise disjoint edges, and, in addition, admits an independent set that consists of$q$consecutive vertices on the boundary of the convex hull? We provide a complete answer to this question, determining$f(n,k,q)$for all relevant values of$n,k$and$q$.  Torsten Sch？neborn Mathematics , 2004, Abstract: We introduce a new Winding Number Conjecture'' about maps from the$(d-1)$-skeleton of the$((d+1)(q-1))$-simplex into$\real^d$. This conjecture is equivalent to the Topological Tverberg Theorem. Furthermore, many statements about the Topological Tverberg Theorem transfer to the Winding Number Conjecture, for example all currently proven cases of the Topological Tverberg Theorem as well as Sierksma's conjecture about the number of Tverberg partitions. In the case$d=2$, the Winding Number Conjecture is a statement about complete graphs: It claims that in every image of$K_{3(q-1)+1}$in the plane either$q-1$triangles wind around one vertex or$q-2$triangles wind around the intersection of two edges, where the triangles, edges and vertices are disjoint. We examine which other graphs have this property and find the minimal subgraph of$K_7$having this property.  Gábor Tardos Mathematics , 2011, Abstract: A graph drawn in the plane with straight-line edges is called a geometric graph. If no path of length at most$k$in a geometric graph$G$is self-intersecting we call$Gk$-locally plane. The main result of this paper is a construction of$k$-locally plane graphs with a super-linear number of edges. For the proof we develop randomized thinning procedures for edge-colored bipartite (abstract) graphs that can be applied to other problems as well.  Computer Science , 2015, Abstract: A simple topological graph is a topological graph in which any two edges have at most one common point, which is either their common endpoint or a proper crossing. More generally, in a k-simple topological graph, every pair of edges has at most k common points of this kind. We construct saturated simple and 2-simple graphs with few edges. These are k-simple graphs in which no further edge can be added. We improve the previous upper bounds of Kyn\v{c}l, Pach, Radoi\v{c}i\'c, and T\'oth and show that there are saturated simple graphs on n vertices with only 7n edges and saturated 2-simple graphs on n vertices with 14.5n edges. As a consequence, 14.5n edges is also a new upper bound for k-simple graphs (considering all values of k). We also construct saturated simple and 2-simple graphs that have some vertices with low degree.  Mathematics , 2014, Abstract: Consider a random geometric graph$G(\chi_n, r_n)$, given by connecting two vertices of a Poisson point process$\chi_n$of intensity$n$on the unit torus whenever their distance is smaller than the parameter$r_n$. The model is conditioned on the rare event that the number of edges observed,$|E|$, is greater than$(1 + \delta)\mathbb{E}(|E|)$, for some fixed$\delta >0$. This article proves that upon conditioning, with high probability there exists a ball of diameter$r_n$which contains a clique of at least$\sqrt{2 \delta \mathbb{E}(|E|)}(1 - \epsilon)$vertices, for any$\epsilon >0\$. Intuitively, this region contains all the "excess" edges the graph is forced to contain by the conditioning event, up to lower order corrections. As a consequence of this result, we prove a large deviations principle for the upper tail of the edge count of the random geometric graph.