On the index-r-free sequences over finite cyclic groups

Abstract

Let [Formula: see text] be a finite cyclic group of order [Formula: see text]. Every sequence [Formula: see text] over [Formula: see text] can be written in the form [Formula: see text] where [Formula: see text] and [Formula: see text], and the index [Formula: see text] of [Formula: see text] is defined as the minimum of [Formula: see text] over all [Formula: see text] with [Formula: see text]. Let [Formula: see text] and [Formula: see text] be any fixed integers. We prove that, for every sufficiently large integer [Formula: see text] divisible by [Formula: see text], there exists a sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] having no subsequence [Formula: see text] of index [Formula: see text], which has substantially improved the previous results in this direction.