On Some Numbers Related to Extremal Combinatorial Sum Problems. (3rd March 2014) - Petrassi, D.
Abstract : Let n, d, and r be three integers such that1 ≤ r,   d ≤ n . Chiaselotti (2002) definedγ n, d, r as the minimum number of the nonnegative partial sums with d summands of a sum∑ 1 = 1 n a i ≥ 0, wherea 1, …, a n are n real numbers arbitrarily chosen in such a way that r of them are nonnegative and the remainingn - r are negative. Chiaselotti (2002) and Chiaselotti et al. (2008) determine the values ofγ n, d, r for particular infinite ranges of the integer parameters n, d, and r . In this paper we continue their approach on this problem and we prove the following results: (i)γ ( n, d, r ) ≤ ( r d ) + ( r d - 1 ) for all values of n, d, and r such that(d - 1) / d n - 1 ≤ r ≤ (d - 1) / d n ; (ii)γ d + 2, d, d = d + 1 .