Monochromatic and Zero-Sum Sets of Nondecreasing Modified Diameter

Grynkiewicz, David J.; Sabar, Rasheed

doi:10.37236/1054

Cited by 12 publications

(3 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Subsum Kneser's Theorem can alternatively be derived as a special case of the DeVos-Goddyn-Mohar Theorem [9] [37]. Theorem C, and the more general Theorem A, have found numerous use in problems regarding sequence subsums [5] [15] [18] [19] [23] [25] [27] [28] [29] [30] [31] [32] [33] [34] [36], extending, complementing or resolving questions of established interest [2] [3] [4] [6] [7] [8] [10] [11] [12] [13] [14] [17] [39] [40] [41] [44] [46] [50].…”

Section: Note |φ H (S ′ )| =mentioning

confidence: 99%

Representing Sequence Subsums as Sumsets of Near Equal Sized Sets

Grynkiewicz¹

2019

Preprint

View full text Add to dashboard Cite

For a sequence S of terms from an abelian group G of length |S|, let Σn(S) denote the set of all elements that can be represented as the sum of terms in some n-term subsequence of S. When the subsum set is very small, |Σn(S)| ≤ |S| − n + 1, it is known that the terms of S can be partitioned into n nonempty sets A1, . . . , An ⊆ G such that Σn(S) = A1 + . . . + An. Moreover, if the upper bound is strict, then |Ai \ Z| ≤ 1 for all i, where Z = n i=1 (Ai + H) and H = {g ∈ G : g + Σn(S) = Σn(S)} is the stabilizer of Σn(S). This allows structural results for sumsets to be used to study the subsum set Σn(S) and is one of the two main ways to derive the natural subsum analog of Kneser's Theorem for sumsets. In this paper, we show that such a partitioning can be achieved with sets Ai of as near equal a size as possible, so ⌊ |S| n ⌋ ≤ |Ai| ≤ ⌈ |S| n ⌉ for all i, apart from one highly structured counterexample when |Σn(S)| = |S| − n + 1 with n = 2. The added information of knowing the sets Ai are of near equal size can be of use when applying the aforementioned partitioning result, or when applying sumset results to study Σn(S) (e.g., [20]). We also give an extension increasing the flexibility of the aforementioned partitioning result and prove some stronger results when n ≥ 1 2 |S| is very large.

show abstract

Section: Note |φ H (S ′ )| =mentioning

confidence: 99%

Representing Sequence Subsums as Sumsets of Near Equal Sized Sets

Grynkiewicz¹

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…Theorem 5 (Bollobás et al [5]). Recent work has been done on generalizations of Question 2, as well as investigations into its relationship to a theorem by Erdős, Ginzburg, and Ziv (see [3,4,6,7,8,10] for examples). With possibly the only exceptions being the bounds given in Theorem 4(d) and (f), as well as Theorem 5(b), all recent work in evaluating f (m, r, t) focuses on parameter families with t = 2 and more recently t = 3 [1].…”

Section: Introductionmentioning

confidence: 99%

On Monochromatic Pairs of Nondecreasing Diameters

O'Neal

Schroeder

2019

Electron. J. Combin.

View full text Add to dashboard Cite

Let $n$, $m$, $r$, $t$ be positive integers and $\Delta:[n]\to[r]$. We say $\Delta$ is $(m,r,t)$-permissible if there exist $t$ disjoint $m$-sets $B_1,\dots,B_t$ contained in $[n]$ for which $|\Delta(B_i)|=1$ for each $i=1,2,\dots,t$, $\max(B_i) < \min(B_{i+1})$ for each $i=1,2,\dots,t-1$, and $\max(B_i)-\min(B_i) \leq \max(B_{i+1})-\max(B_{i+1})$ for each $i=1,2,\dots,t-1$. Let $f(m,r,t)$ be the smallest such $n$ so that all colorings $\Delta$ are $(m,r,t)$-permissible. In this paper, we show that $f(2,2,t)=5t-4$.

show abstract

“…, A n to denote the n-setpartition A, so as to avoid confusion with A 1 • • • A n , which equals the sequence S partitioned/factorized by A. It is easily shown (see [4] [18] [19]) that S has an n-setpartition if and only if h(S) ≤ n ≤ |S|, and if such is the case, then S has an n-setpartition with sets of as near equal a size as possible (i.e., ||A i | − |A j || ≤ 1 for all i and j).…”

mentioning

confidence: 99%

Representation of finite abelian group elements by subsequence sums

Grynkiewicz¹,

Marchan²,

Ordaz³

2009

J. Théor. Nombres Bordeaux

View full text Add to dashboard Cite

Let G ∼ = Cn 1 ⊕ · · · ⊕ Cn r be a finite and nontrivial abelian group with n 1 |n 2 |. .. |nr. A conjecture of Hamidoune says that if W = w 1 · · · wn is a sequence of integers, all but at most one relatively prime to |G|, and S is a sequence over G with |S| ≥ |W | + |G| − 1 ≥ |G| + 1, the maximum multiplicity of S at most |W |, and σ(W) ≡ 0 mod |G|, then there exists a nontrivial subgroup H such that every element g ∈ H can be represented as a weighted subsequence sum of the form g = n P i=1 w i s i , with s 1 · · · sn a subsequence of S. We give two examples showing this does not hold in general, and characterize the counterexamples for large |W | ≥ 1 2 |G|. A theorem of Gao, generalizing an older result of Olson, says that if G is a finite abelian group, and S is a sequence over G with |S| ≥ |G| + D(G) − 1, then either every element of G can be represented as a |G|-term subsequence sum from S, or there exists a coset g + H such that all but at most |G/H| − 2 terms of S are from g + H. We establish some very special cases in a weighted analog of this theorem conjectured by Ordaz and Quiroz, and some partial conclusions in the remaining cases, which imply a recent result of Ordaz and Quiroz. This is done, in part, by extending a weighted setpartition theorem of Grynkiewicz, which we then use to also improve the previously mentioned result of Gao by showing that the hypothesis |S| ≥ |G| + D(G) − 1 can be relaxed to |S| ≥ |G| + d * (G), where d * (G) = r P i=1 (n i − 1). We also use this method to derive a variation on Hamidoune's conjecture valid when at least d * (G) of the w i are relatively prime to |G|. 1. Notation We follow the conventions of [9] for notation concerning sequences over an abelian group. For real numbers a, b ∈ R, we set [a, b] = {x ∈ Z | a ≤ x ≤ b}. Throughout, all abelian groups will be written additively. Let G be an abelian group, and let A, B ⊆ G be nonempty subsets. Then A + B = {a + b | a ∈ A, b ∈ B} denotes their sumset. The stabilizer of A is defined as H(A) = {g ∈ G | g + A = A}, and A is called periodic if H(A) = {0}, and aperiodic otherwise. If A is a union of H-cosets (i.e., H ≤ H(A)), then we say A is H-periodic. The order of an element g ∈ G is denoted ord(g), and we use φ H : G → G/H to denote the natural homomorphism. We use gcd(a, b) to denote the greatest common divisor of a, b ∈ Z. 2000 Mathematics Subject Classification. 11B75 (20K01).

show abstract

Monochromatic and Zero-Sum Sets of Nondecreasing Modified Diameter

Cited by 12 publications

References 28 publications

Representing Sequence Subsums as Sumsets of Near Equal Sized Sets

Representing Sequence Subsums as Sumsets of Near Equal Sized Sets

On Monochromatic Pairs of Nondecreasing Diameters

Representation of finite abelian group elements by subsequence sums

Contact Info

Product

Resources

About