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Independent collections of translates of boxes and a conjecture due to Grünbaum. (English) Zbl 0764.52001

A collection \(\{A_ 1,\dots,A_ n\}\) of \(n\) sets is independent if \(\bigcap^ n_{i=1}X_ i\neq\emptyset\) whenever \(X_ i\in\{A_ i,A_ i^ c\}\) for \(i=1,\dots,n\). A box in \(\mathbb{R}^ d\) is a set of the form \(I_ 1\times\dots\times I_ d\) where each \(I_ j\) is a finite closed interval, and the intervals are in linearly independent subspaces.
The main result of the authors is that the maximum number of boxes in an independent collection, all of which are translates of a given box, is \(\left\lfloor{3d\over 2}\right\rfloor\) for \(d\geq 2\). This shows that the conjecture of \(d+1\) by Grünbaum (1975) is false for \(d\geq 4\), and improves on the upper bound of \(2d\) of Rényi et al. (1951) for this special case.

MSC:

52A37 Other problems of combinatorial convexity
05A18 Partitions of sets
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References:

[1] Grünbaum, B. (1975). Venn diagrams and independent families of sets.Math. Mag.48, 12-23. · Zbl 1206.65233 · doi:10.2307/2689288
[2] Marczewski, E. (1947). Indépendance d’ensembles et prolongements de mesures.Colloq. Math.1, 122-132. · Zbl 0038.03503
[3] Naiman, D. Q., and Wynn, H. P. (1991a). Inclusion-exclusion bonferroni identities and inequalities for discrete tube-like problems via Euler characteristics.Ann. of Statist., to appear. · Zbl 0752.62028
[4] Naiman, D. Q., and Wynn, H. P. (1991b). Discrete Tube Theory inRd. Unpublished manuscript.
[5] Rényi, A., Rényi, C., and Surányi, J. (1951). Sur l’independence des domaines simples dans l’espace euclidien àn dimensions.Colloq. Math.2, 130-135. · Zbl 1056.03031
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