1,309 research outputs found

    On some points-and-lines problems and configurations

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    We apply an old method for constructing points-and-lines configurations in the plane to study some recent questions in incidence geometry.Comment: 14 pages, numerous figures of point-and-line configurations; to appear in the Bezdek-50 special issue of Periodica Mathematica Hungaric

    Finite height lamination spaces for surfaces

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    We describe spaces of essential finite height (measured) laminations in a surface SS using a parameter space we call S\mathbb S, an ordered semi-ring. We show that for every finite height essential lamination LL in SS, there is an action of Ο€1(S)\pi_1(S) on an S\mathbb S-tree dual to the lift of LL to the universal cover of SS

    Transversals in a collections of trees

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    Let S\mathcal{S} be a fixed family of graphs on vertex set VV and G\mathcal{G} be a collection of elements in S\mathcal{S}. We investigated the transversal problem of finding the maximum value of ∣G∣|\mathcal{G}| when G\mathcal{G} contains no rainbow elements in S\mathcal{S}. Specifically, we determine the exact values when S\mathcal{S} is a family of stars or a family of trees of the same order nn with nn dividing ∣V∣|V|. Further, all the extremal cases for G\mathcal{G} are characterized.Comment: 16pages,2figure

    Red-blue clique partitions and (1-1)-transversals

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    Motivated by the problem of Gallai on (1βˆ’1)(1-1)-transversals of 22-intervals, it was proved by the authors in 1969 that if the edges of a complete graph KK are colored with red and blue (both colors can appear on an edge) so that there is no monochromatic induced C4C_4 and C5C_5 then the vertices of KK can be partitioned into a red and a blue clique. Aharoni, Berger, Chudnovsky and Ziani recently strengthened this by showing that it is enough to assume that there is no induced monochromatic C4C_4 and there is no induced C5C_5 in {\em one of the colors}. Here this is strengthened further, it is enough to assume that there is no monochromatic induced C4C_4 and there is no K5K_5 on which both color classes induce a C5C_5. We also answer a question of Kaiser and Rabinovich, giving an example of six 22-convex sets in the plane such that any three intersect but there is no (1βˆ’1)(1-1)-transversal for them
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