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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On Bohman’s conjecture related to a sum packing problem of Erdos
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by R. Ahlswede, H. Aydinian and L. H. Khachatrian PDF
Proc. Amer. Math. Soc. 132 (2004), 1257-1265 Request permission

Abstract:

Motivated by a sum packing problem of Erdős, Bohman discussed an extremal geometric problem which seems to have an independent interest. Let $H$ be a hyperplane in $\mathbb R^n$ such that $H\cap \{0,\pm 1\}^n=\{0^n\}$. The problem is to determine \[ f(n)\triangleq \max _H|H\cap \{0,\pm 1,\pm 2\}^n|.\] Bohman (1996) conjectured that \[ f(n)=\frac 12 (1+\sqrt 2)^n+\frac 12 (1-\sqrt 2)^n.\] We show that for some constants $c_1,c_2$ we have $c_1(2,538)^n<f(n)< c_2(2,723)^n$—disproving the conjecture. We also consider a more general question of the estimation of $|H\cap \{0,\pm 1,\dots ,\pm m\}|$, when $H\cap \{0,\pm 1,\dots ,\pm k\}=\{0^n\}$, $m>k>1$.
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Additional Information
  • R. Ahlswede
  • Affiliation: Department of Mathematics, University of Bielefeld, POB 100131, 33501 Bielefeld, Germany
  • Email: ahlswede@mathematik.uni-bielefeld.de
  • H. Aydinian
  • Affiliation: Department of Mathematics, University of Bielefeld, POB 100131, 33501 Bielefeld, Germany
  • Email: ayd@mathematik.uni-bielefeld.de
  • L. H. Khachatrian
  • Affiliation: Department of Mathematics, University of Bielefeld, POB 100131, 33501 Bielefeld, Germany
  • Email: lk@mathematik.uni-bielefeld.de
  • Received by editor(s): October 22, 2001
  • Received by editor(s) in revised form: August 22, 2002, and January 15, 2003
  • Published electronically: September 29, 2003
  • Communicated by: John R. Stembridge
  • © Copyright 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 1257-1265
  • MSC (2000): Primary 11P99; Secondary 05D05
  • DOI: https://doi.org/10.1090/S0002-9939-03-07188-0
  • MathSciNet review: 2053329