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The fourth generalized Davenport constant of $C_5^3$

Sze Chun Yiu

arXiv:2609.04950v1cs.DMmath.CO

TL;DR

The paper asks whether the Freeze–Schmid lower bound 5k + 10 is attained for C_5^3. Using a finite, computer-assisted reduction and exhaustive branch search, it proves D_4(C_5^3)=30 and consequently D_k(C_5^3)=5k + 10 for every k ≥ 2.

  • Problem

    The paper asks whether the Freeze–Schmid lower bound 5k + 10 is attained for every k ≥ 2.

  • Method

    A saturation argument reduces the remaining case to 60 multiplicity patterns and 78 normalized rank/plane branches, which are exhaustively searched.

  • Results

    D_4(C_5^3)=30, and consequently D_k(C_5^3)=5k + 10 for every k ≥ 2; further values include D_3(C_5^3)=25 and s_≤6(C_5^3)=24.

  • Takeaways & Limitations

    The Freeze–Schmid bound is attained by C_5^3 from k=2 onward through a finite verification of the remaining zero-sum case.

  • Takeaways & Limitations

    The computation for D_3(C_5^3)=25 was not fully re-executed or independently repeated, and the paper reports no external checking of individual branches.

Abstract

from arXiv · show

For a finite abelian group $G$ and $k \geq 1$, the generalized Davenport constant $D_k(G)$ is the least $\ell$ such that every sequence over $G$ of length at least $\ell$ has $k$ pairwise disjoint nonempty zero-sum subsequences. A theorem of Freeze and Schmid gives $D_k(C_5^3) \geq 5k+10$ for every $k \geq 2$. We prove the matching upper bound: $D_4(C_5^3)=30$, and hence $D_k(C_5^3)=5k+10$ for every $k \geq 2$, so the Freeze--Schmid bound is attained by $C_5^3$ from $k=2$ onward, as it is by $C_2^3$ and unlike $C_3^3$. The proof is finite and computer-assisted. The remaining case reduces to showing that every zero-sum sequence of length $31$ over $C_5^3$ contains a nonempty zero-sum subsequence of length at most five. A saturation argument confines the multiplicities of a hypothetical counterexample to $\{1,2,4\}$, its support pattern to one of $60$ solutions of two linear equations, and its geometry to one of $78$ rank/plane branches normalized to a standard basis; an exhaustive search exhausts every branch with no survivor. The search was carried out by three independently written implementations, and the branch cover was regenerated by separate programs from the lemmas alone; two further machine-verified values, $D_3(C_5^3)=25$ and $s_{\leq 6}(C_5^3)=24$, enter the second statement, and their records accompany the paper.

1 Introduction

The paper resolves whether the Freeze–Schmid lower bound is attained for C_5^3, proving exact generalized Davenport constants through a finite computer-assisted reduction. It also situates this result among known elementary p-groups and reports supporting machine-verified values.

  • Reduction: Every zero-sum sequence of length 31 over C_5^3 contains a nonempty zero-sum subsequence of length at most five.This is the finite case that establishes the required upper bound.
  • Main result: 5k + 10 is attained for D_k(C_5^3) for every k ≥ 2, matching the Freeze–Schmid lower bound.The key remaining question was whether the lower bound is attained for every k ≥ 2.
  • Main result: 30 is the exact value of D_4(C_5^3), providing the central upper-bound result.The proof reduces this claim to excluding a specific short-zero-sum-free configuration.
  • Proof strategy: The proof is a finite, computer-assisted case analysis using literature results, structural reductions, and exhaustive computation.The paper states that it claims no new general theory, emphasizing the exact value, finite reduction, and verification path.
  • Supporting computations: D_3(C_5^3) = 25 and s_≤6(C_5^3) = 24 are additional machine-verified values used in the second statement.Their computation records accompany the paper.
  • Context: Among elementary p-groups of rank at least three for p ∈ {2, 3, 5}, the Freeze–Schmid bound is attained for all k ≥ 2 exactly when p ≠ 3.The paper records this as observed data and gives no explanation for the distinction.

2 Notation and ingredients

This section fixes the sequence and zero-sum notation, records characterizations and lemmas for generalized Davenport constants, and states the machine-verified inputs used later. It then reduces the main claims to D_4(C_5^3)=30, D_3(C_5^3)=25, and s_≤6(C_5^3)=24, while documenting computational provenance and limitations.

  • Notation: A short-zero-free sequence has no nonempty zero-sum subsequence of length at most five, with “short” referring to length at most exp(G)=5.The group is identified with (Z/5Z)^3, a vector space over the field with five elements.
  • Notation: D_k(G) is the least length forcing k pairwise disjoint nonempty zero-sum subsequences, and equivalently D_k(G)=max{|B|: B is zero-sum and z(B)≤k}.Here z(B) denotes the maximum number of pairwise disjoint nonempty zero-sum subsequences of B.
  • Elementary lemmas: The two elementary lemmas convert short zero-sum subsequence bounds into generalized Davenport bounds and support the later inductive step.The proof applies these lemmas by adjoining the negative total sum and then analyzing whether a short zero-sum subsequence contains the appended element.
  • Machine-verified inputs: D_2(C_5^3)≥20 is obtained from the general lower bound, while the upper bound uses either Zhao’s lemma or the machine value s_≤7(C_5^3)=19.The machine route applies the lemma relating s_≤7(C_5^3)=19 to D_2(C_5^3).
  • Machine-verified inputs: D_3(C_5^3)=25 is established by combining the D_2 value, a length-six short-zero-sum bound, and exhaustive enumeration of normalized extensions.The computation enumerated 98,622 base sequences and 230,983 extension candidates, finding three disjoint zero-sum subsequences in every tested candidate.
  • Machine-verified inputs: The D_3 computation was not fully independently rerun: a second generator reproduced candidate counts, but the final three-disjoint test was not re-executed on those candidates.An earlier replay terminated before producing a result, and the authors state that they have not repeated the computation.
  • Machine-verified inputs: The archived and recomputed value s_≤6(C_5^3)=24 states that the longest sequence without a nonempty zero-sum subsequence of length at most six has length 23.The computation used normalized rank-three sequences together with direct enumeration in rank at most two.
  • Reduction and consequences: Assuming the machine-verified inputs, Lemma 2.5 gives D_4(C_5^3)=30 and hence D_k(C_5^3)=5k+10 for k=2,3,4.For k≥4, the short-zero-sum lemma and induction extend the formula to all larger k.

3 Structure of a hypothetical counterexample

A hypothetical zero-sum sequence of length 31 without a zero-sum subsequence of length at most five is progressively constrained in multiplicity, support, rank, and projective geometry. These restrictions reduce the possible structures to finitely many patterns and exclude small supports through prior computations.

  • Initial constraints: 31-term counterexamples are assumed zero-sum and short-zero-free, forcing every element to be nonzero with multiplicity at most four.The support initially has at least eight elements because seven elements with multiplicity at most four contribute only 28 terms.
  • Saturation and multiplicities: Sequences with support at least nine are saturated, and every multiplicity is therefore one, two, or four.Saturation and the saturation-defect lemma exclude multiplicity three, while exponent five excludes multiplicities at least five.
  • Multiplicity grammar: For support size s, the multiplicities satisfy a1 + b2 + c4 = s and a1 + 2b2 + 4c4 = 31, yielding exactly 60 patterns for 14 ≤ s ≤ 31.The derived formulas are b2 = 31 − s − 3c4 and a1 = 2s − 31 + 2c4, with a1 odd and at least one.
  • Projective-line restrictions: A projective-line enumeration leaves 21 admissible multiplicity vectors, while multiplicity-four elements are isolated on their lines and high-multiplicity elements are pairwise linearly independent.The line analysis tests all 4^4 = 256 vectors against short zero sums.
  • Rank and support: The support has rank three, every two-dimensional subspace contains at most 12 terms, and the support size is at least 14.The rank conclusion follows because a 31-term counterexample cannot fit inside a two-dimensional subspace.
  • Small-support exclusion: Supports of sizes 8 through 13 are excluded by exhaustive archived computations and exact searches, leaving only supports of size at least 14.Support eight yields 564 normalized extremal sets with no valid completion, while supports nine through 13 have no completion under their remaining patterns.

4 The branch cover

The remaining multiplicity patterns are covered by 78 normalized rank and plane branches. Each branch fixes a pattern, standard-basis seeds, and any plane constraint needed to represent every possible counterexample.

  • Normalization: Every branch normalizes three independent support elements to e1, e2, e3 while retaining the prescribed multiplicity profile and possible plane constraint.The GL(3,5) action preserves lengths, sums, multiplicities, ranks, and short zero sums.
  • Supports 23–31: For supports 23–31, 18 patterns produce 27 branches according to whether the high part has rank three, rank two, one high-multiplicity element, or none.Rank-two branches confine remaining multiplicity-two elements to the plane spanned by the normalized first two basis elements.

5 The exhaustive search

The exhaustive search enumerates each branch’s admissible completions exactly once, rejects short zero sums incrementally, and uses independent implementations and regenerated branch covers to verify completeness.

  • Enumeration: Each branch enumerates remaining elements by multiplicity-four, multiplicity-two, and multiplicity-one stages, with duplicate support points and forbidden projective-line placements skipped.The final multiplicity-one element is forced by the zero-sum condition and accepted only when all admissibility and ordering checks pass.
  • Short-zero detection: Partial sequences are rejected incrementally by tracking attainable sums for every length from zero through five.Adding an element updates each weight set by union with the translated set from the preceding weight.
  • Independent search: Two archived engines and a separately written third engine found no completion across all 78 branches, including 213 plane sub-runs.The third engine rechecked reported completions for test patterns with a separate brute-force routine.
  • Additional values: The same computational framework verified s≤6(C_5^3) = 24, s≤7(C_5^3) = 19, and s≤8(C_5^3) = 18.The T = 6 computation found a rank-three sequence of length 23 but none of length 24.
  • Completeness verification: Separate programs regenerated the 60 patterns and 78 branches from the mathematical lemmas rather than reading the archived branch list or search outputs.The regenerated covers matched the archived cover with no missing or extra branches.

6 Proof of Theorem 1.1(a)

The proof assumes a zero-sum short-zero-free sequence of length 31, reduces it to one of 78 normalized branches, and then uses exhaustive search to rule out every branch.

  • Reduction: A hypothetical length-31 counterexample has support at least 14, is saturated, and has multiplicities only in {1, 2, 4}.Its multiplicity counts therefore satisfy one of the 60 admissible patterns.
  • Branch exhaustion: The branch-cover propositions place an image of every such sequence in one of 78 branch cases.The search then tests whether the branch seeds can be completed to a zero-sum short-zero-free sequence.
  • Contradiction: No branch admits a zero-sum short-zero-free completion, so the assumed length-31 counterexample cannot exist.This closes the finite case analysis used for Theorem 1.1(a).

7 Relation to prior work and scope of the claim

The paper establishes the exact value D4(C_5^3)=30 and extends it to D_k(C_5^3)=5k+10 for every k≥2, while positioning this result relative to prior bounds and known cases. Its finite reduction and verified computation are presented as the main contributions, with reuse limited by the need for a suitable structure theorem.

  • Comparison with prior cases: The value D2(C_5^3)=20 is not regarded as new, since it follows from the Freeze–Schmid bound together with existing results or a machine value.The paper distinguishes this previously obtainable case from the new exact D4 result.
  • Main contribution: D4(C_5^3)=30, with consequence D_k(C_5^3)=5k+10 for all k≥2.This is the exact value and extension identified as the paper’s main addition.
  • Method and scope: The contribution combines a finite reduction with a verified computation that decides the remaining case.The route uses saturation, multiplicity patterns, rank/plane branches normalized to a basis, and exhaustive weight-layered search.
  • Method and scope: The reduction is not specific to the numerical parameters, but Lemma 3.1 requires Property C or another structure theorem at length η(G)−1.This requirement limits immediate reuse to groups where such a theorem is available.
  • Comparison with prior cases: The Freeze–Schmid bound is attained from k=2 onward for C_2^3 and C_5^3, but not for C_3^3.The paper contrasts the new C_5^3 result with the elementary 2-group case and the rank-three ternary case.
  • Open scope: For primes p, the paper leaves open when the Freeze–Schmid bound is attained for all k≥2 and where the threshold in k lies otherwise.The authors explicitly state that this broader question remains unresolved.

8 Provenance of the computations and limits of the claim

The computational claims rely on multiple implementations, replay, and separately regenerated case analyses, but the paper also records important provenance limits. In particular, some inputs have single-implementation provenance, and the programs and branch certificates have not received external review or independent checking.

  • Verification layers: The 78-branch search used three independently written implementations, was replayed on a second machine, and was paired with case analysis regenerated by two separate programs.Archived exclusions of supports 8–13 were also included, with supports 9–13 re-run for this paper.
  • Verification layers: Some machine-verified inputs used in part (b) have single-implementation provenance, while another was recomputed for this paper.The paper distinguishes the provenance of the computational inputs rather than treating them as uniformly replicated.
  • Limitations: No person outside the project has reviewed the case analysis or programs, and no externally produced or externally checked certificates exist for individual branches.The authors defer external review and independent replication to future refereeing or later work.
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