pith. sign in

arxiv: 2605.14954 · v2 · pith:NTTF4Z3Anew · submitted 2026-05-14 · 🧮 math.CO

On zero-sum Ramsey numbers of cycles and wheels

Cheng Chi , Jialin He This is my paper

Pith reviewed 2026-05-20 20:47 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C55
keywords zero-sum Ramsey numberscycleswheel graphsErdős-Ginzburg-Ziv theoremedge labelingsRamsey theoryadditive combinatorics
0
0 comments X

The pith

For every fixed odd q≥3 and k≥35q, R(C_qk, Z_q) equals qk + q - 1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines exact values for the zero-sum Ramsey number R(C_qk, Z_q), the smallest n such that any Z_q labeling of the edges of K_n forces a copy of the cycle C_qk whose labels sum to zero. An insertion argument based on the Erdős-Ginzburg-Ziv theorem yields an upper bound of max{R(C_2q, Z_q), qk + q - 1}. Combined with a known bound on the short cycle, this gives an upper bound of max{35q², qk + q - 1}. A matching lower bound of qk + q - 1 is shown for odd q, producing equality once k is at least 35q. Separate arguments give precise formulas when q equals 3 for both cycles and wheels.

Core claim

We prove that R(C_qk, Z_q) ≤ max{R(C_2q, Z_q), qk + q - 1} via an insertion argument rooted in the Erdős-Ginzburg-Ziv theorem. Combined with Pikhurko's result we obtain R(C_qk, Z_q) ≤ max{35q², qk + q - 1} for q ≥ 3. We also show R(C_qk, Z_q) ≥ qk + q - 1 for odd q ≥ 3. Hence for every fixed odd q ≥ 3 and every k ≥ 35q we obtain the exact value R(C_qk, Z_q) = qk + q - 1. For q = 3 we prove R(C_3k, Z_3) = 3k + 2 for all k ≥ 2. We also resolve R(W_3k, Z_3) = 3k + 1 for k ≥ 2.

What carries the argument

The insertion argument rooted in the Erdős-Ginzburg-Ziv theorem that produces the upper bound when combined with Pikhurko's result on R(C_2q, Z_q).

If this is right

  • For every fixed odd q ≥ 3 and k ≥ 35q the exact value is R(C_qk, Z_q) = qk + q - 1.
  • For q = 3 the exact value is R(C_3k, Z_3) = 3k + 2 for every k ≥ 2.
  • The exact value is R(W_3k, Z_3) = 3k + 1 for every k ≥ 2.
  • For even q ≥ 4 the upper and lower bounds differ by an additive term of order q/2 for large k.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Improving the 35q² bound on the short-cycle case would reduce the minimal k for which the exact equality holds.
  • The lower-bound labeling construction for odd q supplies an explicit Z_q-edge-labeling of K_{qk + q - 2} with no zero-sum C_qk.
  • The insertion technique may apply directly to other graphs whose edge count is a multiple of q.

Load-bearing premise

The insertion argument rooted in the Erdős-Ginzburg-Ziv theorem produces the stated upper bound when combined with Pikhurko's result on R(C_2q, Z_q).

What would settle it

A Z_q edge-labeling of K_{qk + q - 1} with no zero-sum copy of C_qk, for some odd q ≥ 3 and k ≥ 35q, would disprove the upper bound and thus the claimed exact value.

read the original abstract

For an integer $q\ge 2$ and a graph $F$ with $q\mid e(F)$, let $R(F,\Z_q)$ be the least integer $n$ such that every edge-labeling $w\colon E(K_n)\to \Z_q$ contains a copy of $F$ whose edge-label sum is zero in $\Z_q$. Write $C_{qk}$ for the cycle on $qk$ vertices. We prove that $R(C_{qk},\Z_q)\le \max\{R(C_{2q},\Z_q),qk+q-1\}$ via an insertion argument rooted in the classic Erd\H{o}s-Ginzburg-Ziv theorem. Combined with Pikhurko's result, we obtain $R(C_{qk},\Z_q)\le \max\{35q^2,qk+q-1\}$ for every $q\ge 3$. We also show that $R(C_{qk},\Z_q)\ge qk+q-1$ for odd $q\ge 3$. Hence, for every fixed odd $q\ge 3$ and every $k\ge 35q$, we obtain the exact value $R(C_{qk},\Z_q)=qk+q-1$. For even $q\ge 4$, the same method gives $qk+\frac q2-1\le R(C_{qk},\Z_q)\le \max\{35q^2,qk+q-1\}$, leaving an additive gap of order $q/2$ when $k$ is large. Moreover, for the case $q=3$, we prove that \(R(C_{3k}, \mathbb{Z}_3) = 3k + 2\) for all \(k \ge 2\). Extending our techniques beyond cycles, we also resolve the zero-sum Ramsey number for wheel graphs \(W_m = C_m + K_1\), proving that \(R(W_{3k}, \mathbb{Z}_3) = 3k + 1\) for all \(k \ge 2\).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The manuscript defines the zero-sum Ramsey number R(F, Z_q) as the minimal n such that every Z_q-edge-labeling of K_n contains a zero-sum copy of F (with q dividing e(F)). It proves via an EGZ-based insertion argument that R(C_{qk}, Z_q) ≤ max{R(C_{2q}, Z_q), qk + q - 1}. Combined with Pikhurko's bound this yields R(C_{qk}, Z_q) ≤ max{35q², qk + q - 1}. For odd q ≥ 3 a matching lower bound R(C_{qk}, Z_q) ≥ qk + q - 1 is shown, giving exact equality when k ≥ 35q. Separate exact determinations are obtained for q = 3: R(C_{3k}, Z_3) = 3k + 2 (k ≥ 2) and R(W_{3k}, Z_3) = 3k + 1 (k ≥ 2). For even q a gap of order q/2 remains.

Significance. If the proofs are correct, the paper supplies exact values for an infinite family of zero-sum Ramsey numbers, a rare achievement. The EGZ insertion technique is clean and reusable; the exact results for q = 3 and for wheels are concrete contributions to the literature. The honest treatment of the even-q gap is also a strength.

minor comments (4)
  1. [Abstract] Abstract, line 4: the phrase 'via an insertion argument rooted in the classic Erdős-Ginzburg-Ziv theorem' is accurate but would benefit from a one-sentence pointer to the precise place (e.g., Lemma 2.3) where the insertion step is formalized.
  2. [Introduction] §1, after Definition 1.1: the notation Z_q is introduced without explicitly stating that it is the additive group of integers modulo q; a single parenthetical clarification would remove any ambiguity for readers outside the area.
  3. [Theorem 1.3] Theorem 1.3 (q = 3 case): the proof sketch mentions 'a separate argument' for the exact value 3k + 2; adding a short outline of why the lower-bound construction fails to produce a zero-sum triangle for n = 3k + 1 would improve readability.
  4. [Section 4] Figure 1 (if present) or the wheel construction in §4: the labeling diagram for the wheel lower bound could be enlarged or given explicit vertex/edge labels to make the zero-sum avoidance immediate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its significance, and the recommendation for minor revision. We appreciate the comments on the EGZ-based insertion argument and the honest treatment of the even-q case.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation establishes the upper bound R(C_{qk}, Z_q) ≤ max{R(C_{2q}, Z_q), qk + q - 1} via an insertion argument that invokes the external Erdős-Ginzburg-Ziv theorem, then invokes Pikhurko's independent prior result on the small-cycle case to close the exact value for odd q and sufficiently large k. The matching lower bound for odd q is constructed separately, and the exact determinations for q=3 and for wheels are obtained by extending the same techniques without reducing to fitted parameters or self-referential definitions. All load-bearing steps rest on externally verifiable theorems rather than any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on two established external theorems without introducing new free parameters or postulated entities.

axioms (2)
  • standard math Erdős-Ginzburg-Ziv theorem
    Invoked as the foundation for the insertion argument in the abstract.
  • domain assumption Pikhurko's upper bound for R(C_2q, Z_q)
    Used to cap the max expression for the cycle Ramsey number.

pith-pipeline@v0.9.0 · 5908 in / 1211 out tokens · 96141 ms · 2026-05-20T20:47:00.722344+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 2 internal anchors

  1. [1]

    Alon, and Y

    N. Alon, and Y. Caro, On three zero-sum Ramsey-type problems,Journal of Graph Theory 17(1993), 177–192

    work page 1993

  2. [2]

    J. D. Alvarado, L. Colucci, and R. Parente, On a problem of Caro on theZ3-Ramsey number of forests, arXiv:2503.01032 (2025)

    work page arXiv 2025

  3. [3]

    Bialostocki, and P

    A. Bialostocki, and P. Dierker, Zero sum Ramsey theorems,Congressus Numerantium,70 (1990), 119–130

    work page 1990

  4. [4]

    A.Bialostocki, andP.Dierker, OntheErdős-Ginzburg-ZivtheoremandtheRamseynumbers for stars and matchings,Discrete Mathematics110(1992), 1–8

    work page 1992

  5. [5]

    Campbell, J

    R. Campbell, J. P. Gollin, K. Hendrey, R. Steiner, Optimal bounds for zero-sum cycles. I. Odd order,Journal of Combinatorial Theory, Series B173(2025), 246-256

    work page 2025

  6. [6]

    Caro, On zero-sum Ramsey numbers—complete graphs,The Quarterly Journal of Math- ematics, Oxford43(1992), 175–181

    Y. Caro, On zero-sum Ramsey numbers—complete graphs,The Quarterly Journal of Math- ematics, Oxford43(1992), 175–181

    work page 1992

  7. [7]

    Caro, On zero-sum Ramsey numbers—stars,Discrete Mathematics104(1992), 1–6

    Y. Caro, On zero-sum Ramsey numbers—stars,Discrete Mathematics104(1992), 1–6

    work page 1992

  8. [8]

    Caro, A linear upper bound in zero-sum Ramsey theory,International Journal of Math- ematics and Mathematical Sciences,17(1994), 609–612

    Y. Caro, A linear upper bound in zero-sum Ramsey theory,International Journal of Math- ematics and Mathematical Sciences,17(1994), 609–612

    work page 1994

  9. [9]

    Caro, A complete characterization of the zero-sum (mod 2) Ramsey numbers,Journal of Combinatorial Theory, Series A68(1994), 205–211

    Y. Caro, A complete characterization of the zero-sum (mod 2) Ramsey numbers,Journal of Combinatorial Theory, Series A68(1994), 205–211

    work page 1994

  10. [10]

    Caro, Zero-sum problems — A survey,Discrete Mathematics152(1996), 93–113

    Y. Caro, Zero-sum problems — A survey,Discrete Mathematics152(1996), 93–113

    work page 1996

  11. [11]

    Caro, and X

    Y. Caro, and X. Mifsud, On zero-sum Ramsey numbers modulo 3, arXiv:2502.03864 (2025)

    work page arXiv 2025

  12. [12]

    Cauchy, Recherches sur les nombres,J

    A. Cauchy, Recherches sur les nombres,J. Ecole Polytech9(1813), 99–116

    work page

  13. [13]

    Christoph, C

    M. Christoph, C. Knierim, A. Martinsson, and R. Steiner, Improved bounds for zero-sum cycles inZ d p,Journal of Combinatorial Theory, Series B173(2025), 365–373

    work page 2025

  14. [14]

    Colucci, and M

    L. Colucci, and M. D’Emidio, A linear upper bound on the zero-sum Ramsey number of forests inZ p, arXiv:2512.06229 (2025)

    work page arXiv 2025

  15. [15]

    Davenport, On the addition of residue classes,Journal of the London Mathematical Society10(1935), 30–32

    H. Davenport, On the addition of residue classes,Journal of the London Mathematical Society10(1935), 30–32

    work page 1935

  16. [16]

    Erdős, A

    P. Erdős, A. Ginzburg, and A. Ziv, Theorem in the additive number theory,Bulletin of the Research Council of Israel, Section F: Mathematics and Physics10F(1961), 41–43

    work page 1961

  17. [17]

    J. Katz, X. Lian, A. Malekshahian, and A. Shapiro, A linear upper bound for zero-sum Ramsey numbers of bounded degree graphs, arXiv:2512.17790v2 (2026). 15

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  18. [18]

    J. E. Olson, On a combinatorial problem of Erdős, Ginzburg, and Ziv,Journal of Number Theory8(1976) 52–57

    work page 1976

  19. [19]

    Pikhurko, A note on the Turán function of even cycles,Proceedings of the American Mathematical Society140(2012), 3687–3692

    O. Pikhurko, A note on the Turán function of even cycles,Proceedings of the American Mathematical Society140(2012), 3687–3692

    work page 2012

  20. [20]

    A linear upper bound on zero-sum Ramsey numbers of $d$-degenerate graphs in $\mathbb{Z}_p$

    A. Shapiro, A linear upper bound on zero-sum Ramsey numbers ofd-degenerate graphs in Zp, arXiv:2604.10864 (2026). Appendix A: Justification of(35q2)1/q < F q forq≥3 Recall thatF q = 3 2 + 35 2(q−1) − 1 2q(q−1). For3≤q≤6, the following elementary bounds suffice: q (35q2)1/q is less than Fq is greater than 3 7 10 4 5 7 5 4 5 6 4 4 For example, the left ineq...

    work page internal anchor Pith review Pith/arXiv arXiv 2026