On the Olson and the Strong Davenport constants

A subset $S$ of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of $S$ is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, $p$-groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general results, establishing new lower bounds for the cardinality of zero-sumfree sets for various types of groups. The quality of these bounds is explored via the treatment, which is computer-aided, of selected explicit examples. Moreover, we investigate a closely related notion, namely the maximal cardinality of minimal zero-sum sets, i.e., the Strong Davenport constant. In particular, we determine its value for elementary $p$-groups of rank at most $2$, paralleling and building on recent results on this problem for the Olson constant

The Plus-Minus Davenport Constant of Finite Abelian Groups

Let G be a finite abelian group, written additively. The Davenport constant, D(G), is the smallest positive number s such that any subset of the group G, with cardinality at least s, contains a non-trivial zero-subsum. We focus on a variation of the Davenport constant where we allow addition and subtraction in the non-trivial zero-subsum. This constant is called the plus-minus Davenport constant, D±(G). In the early 2000’s, Marchan, Ordaz, and Schmid proved that if the cardinality of G is less than or equal to 100, then the D±(G) is the floor of log2 n + 1, the basic upper bound, with few exceptions. The value of D±(G) is primarily known when the rank of G at most two and the cardinality of G is less than or equal to 100. In most cases, when D±(G) is known, D±(G)= floor(log2 |G|) + 1, with the exceptions of when G is a 3-group or a 5-group. We have studied a class of groups where the cardinality of G is a product of two prime powers. We look more closely to when the primes are 2 and 3, since the plus-minus Davenport constant of a 2-group attains the basic upper bound and while the plus-minus Davenport constant of a 3-group does not. To help us compute D±(G), we define the even plus-minus Davenport constant, De±(G), that guarantees a pm zero-subsum of even length. Let Cn be a cyclic group of order n. Then D(Cn) = n and D±(Cn) =floor( log2 n)+1. We have shown that De±(Cn) depends on whether n is even or odd. When n is even and not a power of 2, then De±(Cn) = floor(log2 n) + 2. When n = 2k , then De±(Cn) = floor(log2 n) + 1. The case when n is odd, De±(Cn) varies depending on how close n is to a power of 2. We have also shown that a subset containing the Jacobsthal numbers provides a subset of Cn that does not contain an even pm zero-subsum for certain values of n. When G is a finite abelian group, we provide bounds for De±(G). If D±(G) is known, then we given an improvement to the lower bound of De±(G). Additional improvements are shown when G is a direct sum an elementary abelian p-groups where p is prime. Then we compute the values of De±(Cr3 ) when 2 ≤ r ≤ 9 and provide an optimal lower bound for larger r. For the group C2 ⊕ Cr3 , D±(C2 ⊕ Cr3 ) = De±(Cr3 ). When r \u3c 10, D±(C2 ⊕ Cr3 ) does not attain the basic upper bound. We conjecture that as r increases, D±(C2 ⊕ Cr3 ) will not attain the basic upper bound. Now, let G = Cq2 ⊕ Cr3 . We compute the values of D±(G) for general q and small r. In this case, we show that if D±(G) attains the basic upper bound then so does De±(G). We then look at the case when the cardinality of G is a product of two prime powers and show improvements on the lower bound by using the fractional part of log2 p of each prime. Furthermore, we compute the values of D±(G) when 100 \u3c |G| ≤ 200, with some exceptions

Remarks on a generalization of the Davenport constant

A generalization of the Davenport constant is investigated. For a finite abelian group $G$ and a positive integer $k$, let $D_k(G)$ denote the smallest $\ell$ such that each sequence over $G$ of length at least $\ell$ has $k$ disjoint non-empty zero-sum subsequences. For general $G$, expanding on known results, upper and lower bounds on these invariants are investigated and it is proved that the sequence $(D_k(G))_{k\in\mathbb{N}}$ is eventually an arithmetic progression with difference $\exp(G)$, and several questions arising from this fact are investigated. For elementary 2-groups, $D_k(G)$ is investigated in detail; in particular, the exact values are determined for groups of rank four and five (for rank at most three they were already known).Comment: Various expository changes, updated and slightly expanded bibliograph

Improved bounds and new techniques for Davenport-Schinzel sequences and their generalizations

Let lambda_s(n) denote the maximum length of a Davenport-Schinzel sequence of order s on n symbols. For s=3 it is known that lambda_3(n) = Theta(n alpha(n)) (Hart and Sharir, 1986). For general s>=4 there are almost-tight upper and lower bounds, both of the form n * 2^poly(alpha(n)) (Agarwal, Sharir, and Shor, 1989). Our first result is an improvement of the upper-bound technique of Agarwal et al. We obtain improved upper bounds for s>=6, which are tight for even s up to lower-order terms in the exponent. More importantly, we also present a new technique for deriving upper bounds for lambda_s(n). With this new technique we: (1) re-derive the upper bound of lambda_3(n) <= 2n alpha(n) + O(n sqrt alpha(n)) (first shown by Klazar, 1999); (2) re-derive our own new upper bounds for general s; and (3) obtain improved upper bounds for the generalized Davenport-Schinzel sequences considered by Adamec, Klazar, and Valtr (1992). Regarding lower bounds, we show that lambda_3(n) >= 2n alpha(n) - O(n), and therefore, the coefficient 2 is tight. We also present a simpler version of the construction of Agarwal, Sharir, and Shor that achieves the known lower bounds for even s>=4.Comment: To appear in Journal of the ACM. 48 pages, 3 figure

Remarks on the plus-minus weighted Davenport constant

For $(G,+)$ a finite abelian group the plus-minus weighted Davenport constant, denoted $\mathsf{D}_{\pm}(G)$, is the smallest $\ell$ such that each sequence $g_1 ... g_{\ell}$ over $G$ has a weighted zero-subsum with weights +1 and -1, i.e., there is a non-empty subset $I \subset \{1,..., \ell\}$ such that $\sum_{i \in I} a_i g_i =0$ for $a_i \in \{+1,-1\}$. We present new bounds for this constant, mainly lower bounds, and also obtain the exact value of this constant for various additional types of groups

Sharp Bounds on Davenport-Schinzel Sequences of Every Order

One of the longest-standing open problems in computational geometry is to bound the lower envelope of $n$ univariate functions, each pair of which crosses at most $s$ times, for some fixed $s$. This problem is known to be equivalent to bounding the length of an order-$s$ Davenport-Schinzel sequence, namely a sequence over an $n$-letter alphabet that avoids alternating subsequences of the form $a \cdots b \cdots a \cdots b \cdots$ with length $s+2$. These sequences were introduced by Davenport and Schinzel in 1965 to model a certain problem in differential equations and have since been applied to bounding the running times of geometric algorithms, data structures, and the combinatorial complexity of geometric arrangements. Let $\lambda_s(n)$ be the maximum length of an order-$s$ DS sequence over $n$ letters. What is $\lambda_s$ asymptotically? This question has been answered satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and Nivasch) when $s$ is even or $s\le 3$. However, since the work of Agarwal, Sharir, and Shor in the mid-1980s there has been a persistent gap in our understanding of the odd orders. In this work we effectively close the problem by establishing sharp bounds on Davenport-Schinzel sequences of every order $s$. Our results reveal that, contrary to one's intuition, $\lambda_s(n)$ behaves essentially like $\lambda_{s-1}(n)$ when $s$ is odd. This refutes conjectures due to Alon et al. (2008) and Nivasch (2010).Comment: A 10-page extended abstract will appear in the Proceedings of the Symposium on Computational Geometry, 201

Upper Bounds for the Davenport Constant

We prove that for all but a certain number of abelian groups of order n its Davenport constant is atmost n/k+k-1 for k=1,2,..,7. For groups of order three we improve on the existing bound involving the Alon-Dubiner constant.Comment: article soumis, decembre 200