Discret. Math. | 2019
On generalized Erdős-Ginzburg-Ziv constants of Cnr
Abstract
Abstract Let G be an additive finite abelian group with exponent exp ( G ) = n . For any positive integer k , the k th Erdős–Ginzburg–Ziv constant s k n ( G ) is defined as the smallest positive integer t such that every sequence S in G of length at least t has a zero-sum subsequence of length k n . It is easy to see that s k n ( C n r ) ≥ ( k + r ) n − r where n , r ∈ N . Kubertin conjectured that the equality holds for any k ≥ r . In this paper, we prove the following results: • [(1)] For every positive integer k ≥ 6 , we have s k n ( C n 3 ) = ( k + 3 ) n + O ( n ln n ) . • [(2)] For every positive integer k ≥ 18 , we have s k n ( C n 4 ) = ( k + 4 ) n + O ( n ln n ) . • [(3)] For n ∈ N , assume that the largest prime power divisor of n is p a for some a ∈ N . Forany fixed r ≥ 5 , if p t ≥ r for some t ∈ N , then for any k ∈ N we have s k p t n ( C n r ) ≤ ( k p t + r ) n + c r n ln n , where c r is a constant that depends on r . Our results verify the conjecture of Kubertin asymptotically in the above cases.