bibkey: lemos2024consecutivedavenport
authors: A. Lemos, A. O. Moura, S. Ribas and A. T. Silva
year: 2024
title: A note on weighted consecutive Davenport constant
doi: 10.48550/arXiv.2404.11312
url: https://arxiv.org/abs/2404.11312v1
claim: 'Conjecture 3.4: For every finite group G, it holds C(G) = |G|.'
strata_touched:
  - D5/S3/FiniteGroups/LemosMouraRibasSilvaConsecutiveDavenport
license: citation-only
triage: anchor

A note on weighted consecutive Davenport constant

A. Lemos, A. O. Moura, S. Ribas and A. T. Silva, “A note on weighted consecutive Davenport constant,” arXiv:2404.11312v1.

Verified locator

DOI: https://doi.org/10.48550/arXiv.2404.11312

Source: https://arxiv.org/abs/2404.11312v1

Source statements

Printed p. 1: “An ordered subsequence T of S is an ordered sequence of the form T = gᵢ · gᵢ₊₁ · … · gⱼ₋₁ · gⱼ for some 1 ≤ i ≤ j ≤ k.” Ordered sequences are lists and ordered subsequences are nonempty contiguous infixes, with multiplication taken in the given order.

Printed p. 2: “When A = {1}, we call S a consecutive product-one sequence and a consecutive product-one free sequence, respectively.”

Printed p. 2: “Let C_A(G) be the smallest positive integer ℓ such that every sequence over G, with length at least ℓ, has an A-weighted consecutive product-one subsequence.” “For A = {1}, we omit the index A.” The unweighted constant is the natural-number infimum of the positive thresholds.

Printed p. 3, Conjecture 3.4: “For every finite group G, it holds C(G) = |G|.”

Scope

The formal statement covers every finite group, including the trivial group. Its proof combines the finite prefix-product pigeonhole principle with the quotient word of an enumeration of the group. The source’s abelian and metacyclic cases and its indicated direct products are within this scope. Larger weight sets are outside this statement. The inverse characterization and extremal-word count are not separately formalized here.

The literature screen in preregistration issue #19765 located no universal finite-group settlement in its checked sources. Completeness of that screen is ASSUMED-UNVERIFIED; no bibliographic-priority claim is made.