A note on a Conjecture of Gao and Zhuang for groups of order 2727

The small Davenport constant ${\mathsf{d}}(G)$ of a finite group $G$ is defined to be the maximal length of a sequence over $G$ which has no non-trivial product-one subsequence. In this paper, we prove that ${\mathsf{d}}(G) = 6$ for the non-abelian group of order $27$ and exponent $3$ and thereby establish a conjecture by Gao and Zhuang for this group

The asymptotic number of score sequences

A tournament on a graph is an orientation of its edges. The score sequence lists the in-degrees in non-decreasing order. Works by Winston and Kleitman (1983) and Kim and Pittel (2000) showed that the number $S_n$ of score sequences on the complete graph $K_n$ satisfies $S_n=\Theta(4^n/n^{5/2})$. In this work, by combining recent combinatorial developments related to score sequences with the limit theory for discrete infinitely divisible distributions, we observe that $n^{5/2}S_n/4^n\to 0.392\ldots$, as conjectured by Tak\'acs (1986).Comment: v2: numerical details adde

Weighted zero-sum problems over C₃ʳ

Let Cn be the cyclic group of order n and set sA(Crn) as the smallest integer ℓ such that every sequence S in Cʳn of length at least ℓ has an A-zero-sum subsequence of length equal to exp(Cʳn), for A = {−1, 1}. In this paper, among other things, we give estimates for sA(C₃ʳ), and prove that sA(C₃³) = 9, sA(C₃⁴) = 21 and 41 ≤ sA(C₃⁵) ≤ 45

On Zero-Sum Rado Numbers for the Equation ax_1 + x_2 = x_3

For every positive integer $a$, let $n = R_{ZS}(a)$ be the least integer, provided it exists, such that for every coloring $$\Delta : \{1, 2, ..., n\} \rightarrow \{0, 1, 2\},$$ there exist three integers $x_1, x_2, x_3$ (not necessarily distinct) such that $$\Delta(x_1) + \Delta(x_2) + \Delta(x_3) \equiv 0\ (mod\ 3)$$ and $$ax_1 +x_2 = x_3.$$ If such an integer does not exist, then $R_{ZS}(a) = \infty.$ The main results of this paper are $$R_{ZS}(2) = 12$$ and a lower bound is found for $R_{ZS}(a)$ where $a \geq 2$

Sequences with small subsum sets

AbstractA conjecture of Gao and Leader, recently proved by Sun, states that if X=(xi)i=1n is a sequence of length n in a finite abelian group of exponent n, then either some subsequence of X sums to zero or the set of all sums of subsequences of X has cardinality at least 2n−1. This conjecture turns out to be a simple consequence of a theorem of Olson and White; we investigate generalizations that are not implied by this theorem. In particular, we prove the following result: if X=(xi)i=1n is a sequence of length n, the terms of which generate a finite abelian group of rank at least 3, then either some subsequence of X sums to zero or the set of all sums of subsequences of X has cardinality at least 4n−5

Sequências de soma zero em algumas famílias de grupos abelianos finitos

Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2018.Neste trabalho apresentamos um resultado para grupos da forma G= , com (exp(H),exp(K))=1. Provamos sob certas hipóteses que, se para todas sequências T em F(G) de tamanho constante |T|=α, com soma zero na componente H e soma constante em K tem-se que |supp(ψ(T))|=1, onde ψ representa a função projeção de G em K. Fazemos uma classificação para a estrutura de todas as sequências de G’=C32 de tamanho s(G’)- 1 que não possuem subsequências de tamanho exp(G’) e soma zero. Dado o grupo abeliano finito de posto quatro, G= ,onde H=C24 e K=C32, com o resultado anterior tem-se: 29 ≤ s( ) ≤ 31. Também apresentamos o valor exato para s(G), onde G= , com H=C23 e K=C32, mais precisamente, s(G)=25. Por fim melhoramos a cota superior da família de grupos abelianos G= , com H=C32 , K=Cn, (n,3)=1 e n ≥ 7. Obtemos que s(G) ≤ 6n +12.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) e Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).In this work we present a result for groups of the form , with (exp(H),exp(K))=1. We prove under certain hipotheses that, if for all sequence T in F(G) of constant length |T|=α, with sum equal to zero in the component H and constant sum in the component K we have |supp(ψ(T))|=1, where ψ represents the projection function from G to K. We obtain a classification for the structure of all the sequences of G’=C32 of length s(G’)- 1 that do not have subsequences of length exp(G’) and sum equal to zero. Given the finite abelian group of rank four G= , with H=C24 and K=C32, using the previous result we have: 29 ≤ s( ) ≤ 31. We also present the exact value of s(G), where G= , with H=C23 and K=C32, more precisely, s(G)=25. Finally we improve the upper bound of the family of abelian groups G= , with H=C32 and K=Cn, with (n,3)=1 and n ≥ 7. We obtain that s(G) ≤ 6n +12

2011 IMSAloquium, Student Investigation Showcase

Inquiry Without Boundaries reflects our students’ infinite possibilities to explore their unique passions, develop new interests, and collaborate with experts around the globe.https://digitalcommons.imsa.edu/archives_sir/1003/thumbnail.jp