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Bulletin of the London Mathematical Society Advance Access published online on June 14, 2009
Bulletin of the London Mathematical Society, doi:10.1112/blms/bdp024
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© 2009 London Mathematical Society

Morris's pigeonhole principle and the Helly theorem for unions of convex sets

Jürgen Eckhoff

Fachbereich Mathematik
Universität Dortmund
D-44221 Dortmund
Germany

Klaus-Peter Nischke

Pfahlweg 2
D-65307 Bad Schwalbach-Lindschied
Germany
klaus-peter.nischke@uni-dortmund.de

Received 1 March 2008. Revision received 16 December 2008.

In 1973, H. C. Morris devised a combinatorial scheme, a ‘generalized pigeonhole principle’, which he used to prove a conjecture of Grünbaum and Motzkin from 1961. This conjecture proposed, in an abstract setting, a Helly-type theorem for certain families of disjoint unions of sets. A geometric instance dealing with disjoint unions of convex sets in Rd was proved in a special case by Larman in 1968 and in the general case by Amenta in 1996. Also covered by the conjecture is a topological extension of Amenta's theorem obtained by Kalai and Meshulam in 2008.

Morris's proof of the generalized pigeonhole principle is extremely involved, and the validity of some of his arguments is open to dispute. In the present paper, the principle is placed on a sound basis and established in a relatively short and transparent manner. This includes a particular case, left open by Morris, which is applied here to families of disjoint unions of boxes in Rd.

2000 Mathematics Subject Classification 52A35 (primary), 05A18, 52A01 (secondary).


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