Graphs and Combinatorics

, Volume 3, Issue 1, pp 251–254 | Cite as

The maximum number of balancing sets

  • Zoltán Füredi


Leta1, ...,a n be a sequence of nonzero real numbers with sum zero.A subsetB of {1, 2,...,n} is called a balancing set if∑ a b = 0 (b ∈ B). S. Nabeya showed that the number of balancing sets is bounded above by\(\left( {\begin{array}{*{20}c} n \\ {{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} \\ \end{array} } \right)\) and this bound achieved forn even witha j =(−1) j . Here his conjecture is verified, showing a tight upper bound\(\left( {\begin{array}{*{20}c} {2k} \\ {k - 1} \\ \end{array} } \right)\) whenn = 2k + 1. The essentially unique extremal configuration is:a1 = 2,a2 = ... =a k = 1,a k+1 = ... =a 2k+1 = -1.


Real Number Nonzero Real Number Extremal Configuration 
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Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Zoltán Füredi
    • 1
    • 2
  1. 1.Department of MathematicsM.I.T.CambridgeUSA
  2. 2.AT&T Bell LaboratoriesMurray HillUSA

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