On zero-sum subsequences of prescribed length

Abstract

Let [Formula: see text] be an additive finite abelian group with exponent [Formula: see text]. For any positive integer [Formula: see text], let [Formula: see text] be the smallest positive integer [Formula: see text] such that every sequence [Formula: see text] in [Formula: see text] of length at least [Formula: see text] has a zero-sum subsequence of length [Formula: see text]. Let [Formula: see text] be the Davenport constant of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite abelian [Formula: see text]-group with [Formula: see text] then [Formula: see text] for every [Formula: see text], which confirms a conjecture by Gao et al. recently, where [Formula: see text] is a prime.

On generalized Erdős-Ginzburg-Ziv constants of $C_n^r$

“…According to the results in [13,19,22], Conjecture 1.1 has been verified for r ≤ 4 except for some cases when p is rather small (p ≤ 3). Recently, Sidorenko [23,24] verified Conjecture 1.1 for C r 2 .…”

“…In this section, we shall prove our main results, Theorem 1.2. Firstly, we have to verify Conjecture 1.1 for some small primes which are the remaining cases in [13,19,22]. Lemma 3.1.…”

“…For finite abelian groups G of rank two, l(G) = 2 (see [11]). Let p be a prime and q a power of p, the above conjecture was verified for C r q where 1 ≤ r ≤ 4 (also more generally for abelian p-group G with D(G) ≤ 4m) except for some cases when p is rather small, see [13,19,22]. For the studies of l(G) for the general cases, we refer to [11,20,22].…”

Extremal Problems on the Hypercube and the Codegree Turán Density of Complete $r$-Graphs

“…contains no zero-sum subsequence of length r.) Obviously, s exp(G) (G) = s(G). Constants s r (G) were studied in [2,10,11,12,13,15,18]. In the case when k is a power of a prime, Gao proved [11,18]) and conjectured that…”

Modified Erdős-Ginzburg-Ziv constants for $\mathbb{Z}_2^d$

“…The generalized Erdős-Ginzburg-Ziv constant s r (G) is the smallest integer s such that every sequence of length s over G has a zero-sum subsequence of length r. If r = exp(G), then s(G) = s exp(G) (G) is the classical Erdős-Ginzburg-Ziv constant. The constants s r (G) have been studied extensively, see for example [4][5][6][7][8][9][10]12]. The following variation of these constants was introduced in [1] and further studied in [2,3,11].…”