Decomposition of Multiple Packings with Subquadratic Union Complexity

8 years 18 days ago

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Let k be a positive integer and let X be a k-fold packing of inﬁnitely many simply connected compact sets in the plane, that is, assume that every point belongs to at most k sets. Suppose that there is a function f(n) = o(n2 ) with the property that any n members of X surround at most f(n) holes, which means that the complement of their union has at most f(n) connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that X can be decomposed into at most p packings, where p is a constant depending only on k and f.

János Pach, Bartosz Walczak

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