Distinct Volume Subsets

Distinct Volume Subsets

Suppose that a and d are positive integers with a ≥ 2. Let h[subscript a,d](n) be the largest integer t such that any set of n points in R[superscript d] contains a subset of t points for which all the nonzero volumes of the ([t over a]) subsets of order a are distinct. Beginning with Erdos in 1957,...

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Bibliographic Details
Main Authors: Conlon, David (Author), Fox, Jacob (Contributor), Gasarch, William (Author), Harris, David G. (Author), Ulrich, Douglas (Author), Zbarsky, Samuel (Author)
Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
Format: Article
Language:English
Published: Society for Industrial and Applied Mathematics, 2015-06-09T13:02:41Z.
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Summary:Suppose that a and d are positive integers with a ≥ 2. Let h[subscript a,d](n) be the largest integer t such that any set of n points in R[superscript d] contains a subset of t points for which all the nonzero volumes of the ([t over a]) subsets of order a are distinct. Beginning with Erdos in 1957, the function h[subscript 2,d](n) has been closely studied and is known to be at least a power of n. We improve the best known bound for h[subscript 2,d](n) and show that h[subscript a,d](n) is at least a power of n for all a and d.
David & Lucile Packard Foundation (Fellowship)
Simons Foundation (Fellowship)
National Science Foundation (U.S.) (Grant DMS-1069197)
Alfred P. Sloan Foundation (Fellowship)
NEC Corporation (MIT Award)

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