Decomposition of Multiple Packings with Subquadratic Union Complexity

2016 Combinatorics, probability & computing  
Supposekis a positive integer and${\cal X}$is ak-fold packingof the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at mostksets. Suppose there is a functionf(n) =o(n2) with the property that anynmembers of${\cal X}$determine at mostf(n)holes, which means that the complement of their union has at mostf(n) bounded connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that${\cal X}$can be
more » ... ecomposed into at mostp(1-fold) packings, wherepis a constant depending only onkandf.
doi:10.1017/s0963548315000280 fatcat:hb3vxzco3jdhrbncqwc4vtulsa