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Covering with Euclidean boxes. (English) Zbl 0623.05041

The authors prove the following box cover theorem: For each dimension d there exists a constant c depending only on d such that any finite (or compact) set \(V\subset {\mathbb{R}}^ d\) contains a subset S with \(| S| \leq c\) and satisfying \(V\subset \cup_{p,q\in S}box(p,q)\) (box(p,q) denotes the smallest box containing p and q and whose faces are parallel to the coordinate axes of \({\mathbb{R}}^ d)\). For \(d=2\), the result was shown, with \(c=4\), by A. Gyárfás and the second author [Combinatorica 3, 351-358 (1983; Zbl 0534.05052)].
As a first consequence of the box cover theorem, it is shown that if H is a d-dimensional box-hypergraph then \(\rho\) (H)\(\leq c_ 0\cdot \alpha (H)^ c\), where \(\alpha\) (H) and \(\rho\) (H) denote, respectively, the stability and the covering number of H, and c and \(c_ 0\) are constants depending only on d. Another consequence is the extension to the general case of boxes defined by arbitrary finite-cone-subdivisions of \({\mathbb{R}}^ d\). Geometric variants of these results, with boxes replaced by angles, are also derived.
Reviewer: J.Weinstein

MSC:

05C65 Hypergraphs
52A37 Other problems of combinatorial convexity
51M20 Polyhedra and polytopes; regular figures, division of spaces

Citations:

Zbl 0534.05052
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References:

[1] I. Bárány, An extension of the Erdös-Szekeres theorem on large angles, to appear.
[2] Berge, C., ()
[3] Chvatal, V., Monochromatic paths in edge-colored graphs, J. combin. theory soc. B,, 13, 69-70, (1972) · Zbl 0221.05064
[4] Cochand, M.; Duchet, P., Sous LES paves, Annals discr. math., 17, 191-202, (1983) · Zbl 0522.52001
[5] Erdös, P.; Szekeres, G., A combinatorial problem in geometry, Compositio math., 2, 463-470, (1935) · Zbl 0012.27010
[6] Gyárfás, A.; Lehel, J., A Helly-type problem in trees, (), 571-584
[7] Gyárfás, A.; Lehel, J., A Ramsey-type problem in directed and bipartite graphs, Periodica math. hung., 3, 299-304, (1973) · Zbl 0267.05115
[8] Gyárfás, A., J. Lehel, Hypergraph families with bounded edge cover or transversal number. combinatorica, 3, 351-358, (1983)
[9] Sands, B.; Sauer, N.; Woodrow, R., On monochromatic paths in edge-coloured diagraphs, J. combin. theory. ser. B, 33, 271-275, (1982) · Zbl 0488.05036
[10] Todd, M.J., ()
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