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The Upper-Bound Theorem for Families of Boxes in ℝd
Published online by Cambridge University Press: 21 December 2009
Abstract
Let be a family of axis-aligned parallelotopes, or boxes, in ℝd. Denote by fk () the number of subfamilies of of size k + 1 with non-empty intersection. In an earlier paper, the author proved that, if f0 () = n and fr() = 0, then fk() ≤ fk(n, d, r) for k = 1,…,r − 1, where fk(n, d, r) is some explicitly given number. The result is best possible for all k. Here it is shown that, if equality is attained for some such k, then equality is attained for each such k.
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- Research Article
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- Copyright © University College London 2007