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arxiv: 2604.11937 · v2 · submitted 2026-04-13 · 🧮 math.CO

On the Ramsey numbers of wheels, cycles, and stars

Louis DeBiasio , Tucker Wimbish This is my paper

Pith reviewed 2026-05-10 15:07 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C55
keywords Ramsey numberswheel graphseven wheelsstarscyclesasymptoticsgraph theory
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The pith

The Ramsey number of even wheels satisfies 5n minus a small parity correction up to 8n plus 664, with asymptotic determinations of star-versus-wheel and cycle-versus-wheel Ramsey numbers for large parameters.

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

Ramsey numbers measure the smallest complete graph size that forces a monochromatic copy of a target graph in any 2-edge-coloring. The paper establishes improved linear bounds specifically for even wheels and derives them from two general asymptotic results: one pairing stars against even wheels and another pairing even cycles against even wheels. These general results settle the behavior for all sufficiently large values of the relevant parameters, cases that had remained open. The same techniques yield a refined linear upper bound for the Ramsey numbers of odd wheels. The overall effect is to pin down the growth rate of these particular Ramsey numbers more precisely than before.

Core claim

For every integer n at least 2 the Ramsey number of the even wheel satisfies 5n minus (1 plus negative one to the n) over 2 is at most R of W sub 2n which is at most 8n plus 664. This follows from two general theorems that asymptotically determine the Ramsey number of a star versus an even wheel and the Ramsey number of an even cycle versus an even wheel whenever both parameters are sufficiently large. The same methods also produce an upper bound of 10n plus a constant for the Ramsey numbers of odd wheels.

What carries the argument

Two general Ramsey-number results, one for stars against even wheels and one for even cycles against even wheels, that serve as the source from which the concrete bounds on wheel Ramsey numbers are extracted.

If this is right

  • The lower bound 5n minus a parity term holds for every even wheel and improves the earlier 4n plus 1 estimate.
  • The upper bound 8n plus 664 holds for every even wheel and improves the earlier 12n minus 2 estimate.
  • The Ramsey number of a star versus an even wheel is asymptotically determined once both sizes are large.
  • The Ramsey number of an even cycle versus an even wheel is asymptotically determined once both sizes are large.
  • The Ramsey number of an odd wheel is at most 10n plus a constant.

Where Pith is reading between the lines

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

  • The linear growth established here suggests that similar ratio-based arguments could be tested on other families such as paths or trees against wheels.
  • Refinements of the proof that produced the additive constant 664 might lower that constant without changing the leading coefficient.
  • Knowing the exact asymptotic for the mixed star-wheel and cycle-wheel cases supplies a template for attacking other bipartite-versus-nonbipartite Ramsey problems.
  • The parity correction in the lower bound indicates that the exact value of R(W sub 2n) may alternate in a simple pattern depending on the parity of n.

Load-bearing premise

The asymptotic statements for stars versus even wheels and even cycles versus even wheels hold once the parameters are large enough, while the explicit upper bound with the additive 664 is proved directly for every n at least 2.

What would settle it

An explicit 2-edge-coloring of the complete graph on 8n plus 663 vertices that contains no monochromatic copy of the even wheel W sub 2n would disprove the stated upper bound.

Figures

Figures reproduced from arXiv: 2604.11937 by Louis DeBiasio, Tucker Wimbish.

Figure 1
Figure 1. Figure 1: The asymptotically exact value of R(C2m, W2n) for all sufficiently large m, n. For the purposes of this figure, we treat m as a function of n. 1.1.3 Cycles versus stars, fans, and wheels As mentioned before, we have K1,2n ⊆ Fn ⊆ W2n and thus R(C2m, K1,2n) ≤ R(C2m, Fn) ≤ R(C2m, W2n). (5) Using Szemer´edi’s regularity lemma, You and Lin [33] proved R(C2m, Fn) = (4 + o(1))m for m ≥ n and R(C2m, Fn) = (2 + o(1… view at source ↗
Figure 2
Figure 2. Figure 2: The asymptotically exact value of R(C2m, W2n) compared to the exact value of R(C2m, K1,2n) for all sufficiently large m, n. So for all sufficiently large m, n we have that R(C2m, Fn) and R(C2m, W2n) are asymptotically equal, and for all sufficiently large 2 ≤ m ≤ n 2 we have that R(C2m, K1,2n) and R(C2m, W2n) are asymptotically equal. This latter result is quite surprising given that W2n is a far more comp… view at source ↗
Figure 3
Figure 3. Figure 3: Proving that there is a copy of W2n having its center in Dt . Note that for all 1 ≤ i < j ≤ t, we have all possible edges between Vi and Vj in the original graph G∗ . Proof. For all i, j ∈ [t] (not necessarily distinct), let Xi,j = {x ∈ Vi : dG(x, Vj ) < dG∗ (x, Vj ) − η|Vj |}. Since (1 − d)eG∗ (Vi , Vj ) ≤ eG(Vi , Vj ) < X x∈Xi,j (dG∗ (x, Vj ) − η|Vj |) + X v∈Vi\Xi,j dG∗ (v, Vj ) = eG∗ (Vi , Vj ) − |Xi,j … view at source ↗
read the original abstract

The wheel $W_{k}$ is the graph on $k+1$ vertices consisting of a vertex joined to a cycle of length $k$, and we say that $W_k$ is an even wheel if $k$ is even. Mao, Wang, Magnant, Schiermeyer proved that the Ramsey number of $W_{2n}$ is between $4n+1$ and $12n-2$. We improve both of these bounds, showing that $5n-\frac{1+(-1)^{n}}{2}\leq R(W_{2n})\leq 8n+664$ for all integers $n\geq 2$. The main focus of the paper concerns two general results on the Ramsey numbers of stars versus even wheels and even cycles versus even wheels, from which the above bounds are obtained as a corollary. That is, we asymptotically determine $R(K_{1,m}, W_{2n})$ and $R(C_{2m}, W_{2n})$ for all sufficiently large $m$ and $n$, both of which were open problems for most regimes. As for odd wheels, we note that the analogous values for stars versus odd wheels and odd cycles versus odd wheels were already known exactly, from which it follows that $6n+4=R(K_{1,2n+1}, W_{2n+1})\leq R(W_{2n+1})\leq 2\cdot R(C_{2n+1}, W_{2n+1})=12n+2$. Very recently, Zhang and Chen improved the upper bound to $R(W_{2n+1})\leq \frac{32n}{3}+O(1)$. We are able to refine their proof, further improving the upper bound to $R(W_{2n+1})\leq 10n+O(1)$.

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

1 major / 3 minor

Summary. The manuscript claims improved bounds on the Ramsey number of even wheels: 5n - (1 + (-1)^n)/2 ≤ R(W_{2n}) ≤ 8n + 664 for all integers n ≥ 2. The core technical content consists of two general theorems asymptotically determining R(K_{1,m}, W_{2n}) and R(C_{2m}, W_{2n}) for all sufficiently large m and n; the wheel bounds are derived as corollaries. The paper also refines the upper bound for odd wheels to R(W_{2n+1}) ≤ 10n + O(1), improving on recent work by Zhang and Chen.

Significance. If the general theorems hold, the results are significant because they asymptotically determine two families of mixed Ramsey numbers (stars vs. even wheels, cycles vs. even wheels) that were previously open in most regimes. The explicit bounds for R(W_{2n}) tighten both ends of the prior interval [4n+1, 12n-2]. The work uses standard Ramsey-theoretic techniques and supplies concrete, checkable improvements.

major comments (1)
  1. [Corollary following Theorem on cycles vs. even wheels] The upper bound R(W_{2n}) ≤ 8n + 664 for all n ≥ 2 is obtained in the corollary following the general theorem on R(C_{2m}, W_{2n}). The general theorem is stated only for sufficiently large m and n, yet the corollary applies it directly for every n ≥ 2. The manuscript must either specify the explicit threshold beyond which the general result holds and verify the bound for all smaller n by direct computation or case analysis, or show that the proof technique yields the stated constant uniformly for n ≥ 2.
minor comments (3)
  1. [Abstract] In the abstract the lower-bound expression 5n - (1 + (-1)^n)/2 is compact but opaque; spelling out the two cases (5n-1 when n even, 5n when n odd) would improve immediate readability without changing the mathematics.
  2. [Introduction] The introduction compares the new bounds to Mao et al. but would benefit from a small table listing the previous lower/upper estimates alongside the new ones for quick reference.
  3. [Section on odd wheels] The refinement of Zhang and Chen's argument for odd wheels is summarized in one paragraph; a short outline of the key modifications (e.g., which lemma is strengthened) would help readers assess the improvement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript and the recommendation for minor revision. The single major comment identifies a valid issue with the transition from the general asymptotic theorem to the explicit corollary for all n ≥ 2. We address it directly below.

read point-by-point responses

  1. Referee: [Corollary following Theorem on cycles vs. even wheels] The upper bound R(W_{2n}) ≤ 8n + 664 for all n ≥ 2 is obtained in the corollary following the general theorem on R(C_{2m}, W_{2n}). The general theorem is stated only for sufficiently large m and n, yet the corollary applies it directly for every n ≥ 2. The manuscript must either specify the explicit threshold beyond which the general result holds and verify the bound for all smaller n by direct computation or case analysis, or show that the proof technique yields the stated constant uniformly for n ≥ 2.

    Authors: We agree that the current presentation leaves a gap in rigor. The proof of the general theorem on R(C_{2m}, W_{2n}) uses standard Ramsey-theoretic techniques (including the regularity lemma and careful counting arguments) whose quantitative bounds depend on m and n in a way that the additive constant 664 can be made to work for every n ≥ 2 once m is chosen sufficiently large relative to n. Nevertheless, the manuscript does not currently state an explicit threshold or perform the small-n verification. In the revised version we will (i) add a precise statement of the threshold (which the proof shows is n ≥ 2 with m ≥ m_0(n) for an explicit, computable m_0), (ii) verify the bound R(W_{2n}) ≤ 8n + 664 directly for all n from 2 up to a small explicit N by appealing to known small Ramsey numbers and exhaustive checks where necessary, and (iii) confirm that the same constant 664 continues to hold for larger n. These additions will be placed immediately after the statement of the corollary. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper derives improved bounds on R(W_{2n}) explicitly as corollaries of two new general theorems establishing the asymptotic values of R(K_{1,m}, W_{2n}) and R(C_{2m}, W_{2n}) for sufficiently large m and n. These theorems are proved from first principles using standard Ramsey-theoretic arguments (coloring lemmas, embedding techniques, and extremal constructions) that are spelled out in the manuscript without any fitted parameters, self-referential definitions, or reductions to prior results by the same authors. Prior bounds are cited from external work (Mao et al.), and the odd-wheel improvements refine an independent recent result by Zhang and Chen. No step equates a claimed prediction or uniqueness statement to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper builds on standard graph theory and Ramsey theory without introducing new free parameters or invented entities. All contributions consist of new bounds and determinations derived via combinatorial arguments within established frameworks.

axioms (1)
  • standard math Basic properties of graphs, cycles, wheels, and Ramsey numbers
    The work relies on standard definitions and known properties from graph theory and Ramsey theory.

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Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages · 1 internal anchor

  1. [1]

    Allen, T

    P. Allen, T. Luczak, J. Polcyn, and Y. Zhang. The Ramsey number of a long even cycle versus a star.Journal of Combinatorial Theory, Series B, 162:144–153, 2023

    work page 2023

  2. [2]

    J. A. Bondy. Pancyclic graphs I.J. Combin. Theory Ser. B, 11(1):80–84, 1971

    work page 1971

  3. [3]

    Brandt, R

    S. Brandt, R. Faudree, and W. Goddard. Weakly pancyclic graphs.Journal of Graph Theory, 27(3):141–176, 1998

    work page 1998

  4. [4]

    Y. Chen, T. E. Cheng, C. T. Ng, and Y. Zhang. A theorem on cycle–wheel Ramsey number. Discrete Mathematics, 312(5):1059–1061, 2012

    work page 2012

  5. [5]

    DeBiasio and T

    L. DeBiasio and T. Wimbish. On the Ramsey numbers of fans and stars.arXiv preprint arXiv:2603.27110, 2026

    work page arXiv 2026

  6. [6]

    G. A. Dirac. Some theorems on abstract graphs.Proceedings of the London Mathematical Society, 3(1):69–81, 1952

    work page 1952

  7. [7]

    T. Dzido. A note on Tur´ an numbers for even wheels.Graphs and Combinatorics, 29(5):1305–1309, 2013

    work page 2013

  8. [8]

    R. J. Faudree and R. H. Schelp. All Ramsey numbers for cycles in graphs.Discrete Mathematics, 8(4):313–329, 1974

    work page 1974

  9. [9]

    Gould, P

    R. Gould, P. Haxell, and A. Scott. A note on cycle lengths in graphs.Graphs and Combi- natorics, 18:491–498, 2002

    work page 2002

  10. [10]

    Haghi and H

    S. Haghi and H. Maimani. A note on the Ramsey number of even wheels versus stars. Discussiones Mathematicae: Graph Theory, 38(2), 2018

    work page 2018

  11. [11]

    Harborth and I

    H. Harborth and I. Mengersen. All Ramsey numbers for five vertices and seven or eight edges.Discrete Mathematics, 73(1-2):91–98, 1988

    work page 1988

  12. [12]

    Hasmawati

    J. Hasmawati. Bilangan Ramsey untuk graf bintang terhadap graf roda.Tesis Magister, Departemen Matematika ITB, Indonesia, 2004

    work page 2004

  13. [13]

    G. R. Hendry. Ramsey numbers for graphs with five vertices.Journal of Graph Theory, 13(2):245–248, 1989

    work page 1989

  14. [14]

    B. Jackson. Cycles in bipartite graphs.Journal of Combinatorial Theory, Series B, 30(3):332–342, 1981

    work page 1981

  15. [15]

    S. Letzter. An improvement on Luczak’s connected matchings method.Bulletin of the London Mathematical Society, 54(2):609–623, 2022

    work page 2022

  16. [16]

    Li and I

    B. Li and I. Schiermeyer. On star–wheel Ramsey numbers.Graphs and Combinatorics, 32:733–739, 2016

    work page 2016

  17. [17]

    Lidicky and F

    B. Lidicky and F. Pfender. Semidefinite programming and Ramsey numbers.SIAM journal on discrete mathematics, 35(4):2328–2344, 2021

    work page 2021

  18. [18]

    Lin and Y

    Q. Lin and Y. Li. On Ramsey numbers of fans.Discrete applied mathematics, 157(1):191– 194, 2009

    work page 2009

  19. [19]

    Luczak.R(C n, Cn, Cn)≤(4 +o(1))n.Journal of Combinatorial Theory, Series B, 75(2):174–187, 1999

    T. Luczak.R(C n, Cn, Cn)≤(4 +o(1))n.Journal of Combinatorial Theory, Series B, 75(2):174–187, 1999

    work page 1999

  20. [20]

    Y. Mao, Z. Wang, C. Magnant, and I. Schiermeyer. Ramsey and Gallai-Ramsey number for wheels.Graphs and Combinatorics, 38(2):42, 2022. 28

    work page 2022

  21. [21]

    J. W. Moon and L. Moser. On hamiltonian bipartite graphs.Israel Journal of Mathematics, 1(3):163–165, 1963

    work page 1963

  22. [22]

    W. R. Pulleyblank. Fractional matchings and the Edmonds-Gallai theorem.Discrete applied mathematics, 16(1):51–58, 1987

    work page 1987

  23. [23]

    Radziszowski

    S. Radziszowski. Small Ramsey numbers.The Electronic Journal of Combinatorics, Dy- namic Surveys:#DS1: Sep 6, 2024, 1994

    work page 2024

  24. [24]

    Raeisi and A

    G. Raeisi and A. Zaghian. Ramsey number of wheels versus cycles and trees.Canadian Mathematical Bulletin, 59(1):190–196, 2016

    work page 2016

  25. [25]

    V. Rosta. On a Ramsey-type problem of JA Bondy and P Erd˝ os. II.Journal of Combina- torial Theory, Series B, 15(1):105–120, 1973

    work page 1973

  26. [26]

    Schmeichel and S

    E. Schmeichel and S. Hakimi. Pancyclic graphs and a conjecture of Bondy and Chv´ atal. Journal of Combinatorial Theory, Series B, 17(1):22–34, 1974

    work page 1974

  27. [27]

    L. Shi. Ramsey numbers of long cycles versus books or wheels.European Journal of Combinatorics, 31(3):828–838, 2010

    work page 2010

  28. [28]

    Simonovits

    M. Simonovits. A method for solving extremal problems in graph theory, stability problems. InTheory of Graphs (Proc. Colloq., Tihany, 1966), pages 279–319, 1968

    work page 1966

  29. [29]

    Surahmat, E. T. Baskoro, and I. Tomescu. The Ramsey numbers of large cycles versus odd wheels.Graphs and Combinatorics, 24(1):53–58, 2008

    work page 2008

  30. [30]

    Szemer´ edi

    E. Szemer´ edi. Regular partitions of graphs. InProbl` emes combinatoires et th´ eorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), volume 260 ofColloq. Inter- nat. CNRS, pages 399–401. CNRS, Paris, 1978

    work page 1976

  31. [31]

    Van Overberghe

    S. Van Overberghe. Algorithms for computing Ramsey numbers.MS Thesis, Ghent Uni- versity, Belgium, 2020

    work page 2020

  32. [32]

    Voss and C

    H.-J. Voss and C. Zuluaga. Maximale gerade und ungerade Kreise in Graphen. I.Wiss. Z. Tech. Hochsch. Ilmenau, 23(4):57–70, 1977

    work page 1977

  33. [33]

    You and Q

    C. You and Q. Lin. Ramsey numbers of large even cycles and fans.The Electronic Journal of Combinatorics, pages P3–14, 2023

    work page 2023

  34. [34]

    L.-T. Yuan. Extremal graphs for odd wheels.Journal of Graph Theory, 98(4):691–707, 2021

    work page 2021

  35. [35]

    Zhang, H

    Y. Zhang, H. Broersma, and Y. Chen. Three results on cycle-wheel Ramsey numbers. Graphs and Combinatorics, 31(6):2467–2479, 2015

    work page 2015

  36. [36]

    Zhang, H

    Y. Zhang, H. J. Broersma, and Y. Chen. Narrowing down the gap on cycle-star Ramsey numbers.Journal of combinatorics (Somerville), 7(2-3):481–493, 2016

    work page 2016

  37. [37]

    New bounds for Ramsey numbers involving graphs with a center

    Y. Zhang and Y. Chen. New bounds for Ramsey numbers involving graphs with a center. arXiv preprint 2604.13850, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  38. [38]

    Zhang, Y

    Y. Zhang, Y. Zhang, and Y. Chen. The Ramsey numbers of wheels versus odd cycles. Discrete Mathematics, 323:76–80, 2014

    work page 2014

  39. [39]

    Y. Zhao. Graph theory and additive combinatorics: exploring structure and randomness. Cambridge University Press, 2023. 29

    work page 2023