How many atoms can be defined by boxes?

Abstract

We study the functionb(n, d), the maximal number of atoms defined byn d-dimensional boxes, i.e. parallelopipeds in thed-dimensional Euclidean space with sides parallel to the coordinate axes.

We characterize extremal interval families definingb(n, 1)=2n-1 atoms and we show thatb(n, 2)=2n 2-6n+7.

We prove that for everyd,\(b^* (d) = \mathop {\lim }\limits_{n \to \infty } b(n,d)/n^d \) exists and\(1 \leqq (d/2)\sqrt[d]{{b^* (d)}} \leqq e\).

Moreover, we obtainb*(3)=8/9.

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Gyárfás, A., Lehel, J. & Tuza, Z. How many atoms can be defined by boxes?. Combinatorica 5, 193–204 (1985). https://doi.org/10.1007/BF02579362

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AMS subject classification (1980)

  • 51 M 05
  • 52 A 20