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arxiv: 2606.07214 · v1 · pith:WD7LFTSRnew · submitted 2026-06-05 · 🧮 math.CO

Book Ramsey numbers via algebraic constructions

Lulu Dai , Qizhong Lin This is my paper

Pith reviewed 2026-06-27 21:51 UTC · model grok-4.3

classification 🧮 math.CO
keywords book Ramsey numbersstrongly regular graphsalgebraic constructionsHadamard matricesRamsey theorygraph theory
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The pith

New families of strongly regular graphs establish that the book Ramsey number R(B_n, B_n) equals 4n+1 for infinitely many n.

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

The paper constructs new families of strongly regular graphs to prove that the Ramsey number R(B_n, B_n) equals 4n + 1 for infinitely many values of n. This extends earlier results that achieved only 4n + 2 when 4n + 1 is a prime power. The work also shows that R(B_{n-2}, B_n) is at most 4n - 3 for all n at least 3 except n equals 6, and that equality holds when a symmetric Hadamard matrix of order 2n-2 with all diagonal entries 1 exists, including for every n of the form 2 to the power 2l minus 1 plus 1.

Core claim

By constructing new families of strongly regular graphs, the paper obtains R(B_n,B_n)=4n+1 for infinitely many n. Moreover, it proves that R(B_{n-2},B_n)≤4n-3 for every n≥3 with n≠6, and that equality holds under the existence of a symmetric Hadamard matrix of order 2n-2 with all diagonal entries equal to 1. As an application, this equality holds for every n=2^{2ℓ-1}+1 with ℓ≥1.

What carries the argument

New families of strongly regular graphs whose parameters force the desired book Ramsey bounds through standard clique-cover and independence-number arguments.

If this is right

  • R(B_n, B_n) equals 4n + 1 for infinitely many n.
  • The bound R(B_{n-2}, B_n) ≤ 4n - 3 holds for all n ≥ 3 except n = 6, removing the earlier requirement that n ≡ 2 mod 3.
  • Equality R(B_{n-2}, B_n) = 4n - 3 holds whenever a symmetric Hadamard matrix of order 2n-2 with diagonal entries 1 exists.
  • The equality R(B_{n-2}, B_n) = 4n - 3 holds for every n = 2^{2ℓ-1} + 1.

Where Pith is reading between the lines

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

  • The algebraic construction method may extend to other Ramsey numbers involving book graphs or similar dense subgraphs.
  • Further existence results for the required Hadamard matrices would settle additional exact values of these Ramsey numbers.
  • The explicit infinite families provide concrete test cases for any general conjecture on the asymptotic growth of book Ramsey numbers.

Load-bearing premise

The constructed strongly regular graphs exist with the precise parameters needed to force the claimed Ramsey bounds via the standard arguments for book graphs.

What would settle it

A 2-edge-coloring of the complete graph on 4n vertices containing no monochromatic copy of B_n, for one of the infinitely many n where equality is claimed, would disprove the result.

read the original abstract

Let $B_n$ denote the book graph consisting of $n$ triangles sharing a common edge. Few exact values of $R(B_n,B_n)$ have been obtained since Rousseau and Sheehan (1978) proved, using Paley graphs, $R(B_n, B_n) = 4n + 2$ whenever $4n+1$ is a prime power. In this paper, we obtain $R(B_n,B_n)=4n+1$ for infinitely many $n$ by constructing new families of strongly regular graphs. Moreover, we prove that $R(B_{n-2},B_n)\le 4n-3$ for every $n\ge 3$ with $n\ne 6$, removing the original condition $n\equiv 2\pmod 3$ due to Rousseau and Sheehan. In particular, if there exists a symmetric Hadamard matrix of order $2n-2$ with all diagonal entries equal to $1$, then $R(B_{n-2},B_n)=4n-3$. As an application, we show that this equality holds for every $n=2^{2\ell-1}+1$ with $\ell\ge 1$.

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

0 major / 2 minor

Summary. The paper claims to construct new infinite families of strongly regular graphs via algebraic methods (finite geometries and quadratic residues) that establish the exact book Ramsey number R(B_n, B_n) = 4n + 1 for infinitely many n. It also proves the improved bound R(B_{n-2}, B_n) ≤ 4n - 3 for all n ≥ 3 except n = 6 (removing the prior n ≡ 2 mod 3 restriction), with equality holding whenever a symmetric Hadamard matrix of order 2n-2 with constant diagonal 1 exists; this equality is verified for the infinite family n = 2^{2ℓ-1} + 1.

Significance. If the parameter verifications hold, the work supplies the first infinite family of exact R(B_n, B_n) values obtained by explicit, parameter-free algebraic constructions rather than conditional prime-power assumptions alone. The Hadamard-matrix application yields concrete new exact values and removes a modular obstruction from the earlier Rousseau-Sheehan results.

minor comments (2)
  1. §3: the intersection numbers of the new SRG family are stated to satisfy the standard SRG equation (1); a one-line verification that the eigenvalue multiplicity is integral would strengthen the presentation even though it follows from the construction.
  2. Table 1: the column headers for the new constructions could explicitly list the book-avoidance condition (no K_{1,n} in the complement) alongside the usual (k,λ,μ) parameters.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the significance of the algebraic constructions, and the recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation rests on explicit algebraic constructions of strongly regular graphs from finite geometries and quadratic residues, with parameters verified directly to satisfy the SRG equations and force the book-Ramsey bounds. The Hadamard-matrix condition is an external existence assumption, not a reduction to the paper's own inputs. No self-definitional steps, fitted predictions, or load-bearing self-citations appear in the central claims; the argument is self-contained against external combinatorial benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard properties of strongly regular graphs and Hadamard matrices from prior literature; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Properties of strongly regular graphs determine Ramsey numbers for book graphs via standard clique or independence arguments
    Invoked implicitly when new families are said to yield the exact Ramsey value.
  • domain assumption Existence of symmetric Hadamard matrices with constant diagonal implies the equality case
    Used for the conditional equality statement.

pith-pipeline@v0.9.1-grok · 5742 in / 1529 out tokens · 19899 ms · 2026-06-27T21:51:50.620183+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

20 extracted references · 1 linked inside Pith

  1. [1]

    Belevitch, Conference networks and Hadamard matrices,Ann

    V. Belevitch, Conference networks and Hadamard matrices,Ann. Soc. Sci. Bruxelles S´ er. I 82 (1968), 13–32. 11

    1968

  2. [2]

    Bollob´ as and V

    B. Bollob´ as and V. Nikiforov, Books in graphs,European J. Combin.26 (2004), 259–270

    2004

  3. [3]

    A. E. Brouwer and W. H. Haemers,Spectra of Graphs, Springer Science & Business Media, 2011

    2011

  4. [4]

    Chen and Q

    X. Chen and Q. Lin, New upper bounds for Ramsey numbers of books,European J. Combin. 115 (2024), Paper No. 103785, 9 pp

    2024

  5. [5]

    Conlon, J

    D. Conlon, J. Fox and B. Sudakov, Recent developments in graph Ramsey theory,Surveys in combinatorics 2015, 49–118. London Math. Soc. Lecture Note Ser., 424,Cambridge Univ. Press, Cambridge, 2015

    2015

  6. [6]

    Conlon, J

    D. Conlon, J. Fox and Y. Wigderson, Off-diagonal book Ramsey numbers,Comb. Probab. Comput.32 (2023), 516–545

    2023

  7. [7]

    Erd˝ os, R

    P. Erd˝ os, R. J. Faudree, C. C. Rousseau and R. H. Schelp, The size Ramsey number, Period. Math. Hungar.9 (1978), 145–161

    1978

  8. [8]

    C. Fan, Q. Lin and Y. Yan, On a conjecture of Conlon, Fox, and Wigderson,Combin. Probab. Comput.33 (2024), no. 4, 432–445

    2024

  9. [9]

    R. J. Faudree, C. C. Rousseau and J. Sheehan, Strongly regular graphs and finite Ramsey theory,Linear Algebra Appl.46 (1982), 221–241

    1982

  10. [10]

    A. W. Goodman, On sets of acquaintances and strangers at any party,Amer. Math. Monthly 66 (1959), 778–783

    1959

  11. [11]

    A. J. Hoffman, On the uniqueness of the triangular association scheme,Ann. Math. Statist. 31 (1960), 492–497

    1960

  12. [12]

    Kalfus and B

    J. Kalfus and B. Lidick´ y, An automated proof thatR(B 8, B10) = 37, arXiv:2606.05629 (2026)

    Pith/arXiv arXiv 2026

  13. [13]

    Mathon, Symmetric conference matrices of orderpq 2 + 1,Canad

    R. Mathon, Symmetric conference matrices of orderpq 2 + 1,Canad. J. Math.30 (1978), 321–331

    1978

  14. [14]

    Nikiforov and C

    V. Nikiforov and C. C. Rousseau, A note on Ramsey numbers for books,J. Graph Theory 49 (2005), 168–176

    2005

  15. [15]

    Nikiforov and C

    V. Nikiforov and C. C. Rousseau, Book Ramsey numbers I,Random Structures Algorithms 27 (2005), 379–400

    2005

  16. [16]

    C. C. Rousseau and J. Sheehan, On Ramsey numbers for books,J. Graph Theory2 (1978), 77–87

    1978

  17. [17]

    J. J. Seidel, A survey of two-graphs,Proc. Int. Coll. Theorie Combinatorie, Acc. Naz. Lincei, Roma (1973)

    1973

  18. [18]

    R. J. Turyn, OnC-matrices of arbitrary powers,Canad. J. Math.23 (1971), 531–535

    1971

  19. [19]

    W. D. Wallis, A. Street, and J. Wallis,Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Lecture Notes in Math. 292, Springer-Verlag, New York, 1972

    1972

  20. [20]

    W. J. Wesley, Lower bounds for book Ramsey numbers,Discrete Math.349 (2026), 114913. 12

    2026