Graphs and Combinatorics

, Volume 3, Issue 1, pp 251–254

# The maximum number of balancing sets

• Zoltán Füredi
Article

## Abstract

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.

### Keywords

Real Number Nonzero Real Number Extremal Configuration

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