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- W1928602502 abstract "Let X be a convex curve in the plane (say, the unit circle), and let $${mathcal {S}}$$ be a family of planar convex bodies such that every two of them meet at a point of X. Then $${mathcal {S}}$$ has a transversal $$Nsubset {mathbb {R}}^2$$ of size at most $$1.75times 10^9$$ . Suppose instead that $${mathcal {S}}$$ only satisfies the following “(p, 2)-condition”: Among every p elements of $${mathcal {S}}$$ , there are two that meet at a common point of X. Then $${mathcal {S}}$$ has a transversal of size $$O(p^8)$$ . For comparison, the best known bound for the Hadwiger–Debrunner (p, q)-problem in the plane, with $$q=3$$ , is $$O(p^6)$$ . Our result generalizes appropriately for $${mathbb {R}}^d$$ if $$Xsubset {mathbb {R}}^d$$ is, for example, the moment curve." @default.
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- W1928602502 date "2015-08-12" @default.
- W1928602502 modified "2023-09-23" @default.
- W1928602502 title "A Variant of the Hadwiger–Debrunner (p, q)-Problem in the Plane" @default.
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- W1928602502 doi "https://doi.org/10.1007/s00454-015-9723-9" @default.
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