Book Ramsey numbers via algebraic constructions

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 $\ell\ge 1$.

• The paper presents a novel method using pseudo-cyclic strongly regular graphs to achieve R(Bₙ, Bₙ) = 4n+1, extending results beyond prime power cases.
• It removes previous congruence restrictions by applying spectral graph theory to show that R(Bₙ₋₂, Bₙ) ≤ 4n–3 for nearly-diagonal Ramsey numbers.
• The work leverages symmetric conference and Hadamard matrices to construct extremal colorings and generate infinite families of exact book Ramsey numbers.

Algebraic Constructions for Book Ramsey Numbers

Introduction

The paper "Book Ramsey numbers via algebraic constructions" (2606.07214) addresses significant open problems in Ramsey theory concerning the Ramsey numbers of book graphs. For a book graph $B_n$ (consisting of $n$ triangles sharing a common edge), the diagonal Ramsey number $R(B_n,B_n)$ is defined as the smallest integer $N$ such that any red/blue coloring of the edges of $K_N$ contains a monochromatic copy of $B_n$ of either color.

Historically, exact values for $R(B_n, B_n)$ have been scarce, apart from settings where classical Paley graphs yield extremal constructions—namely, when $4n+1$ is a prime power. This work breaks new ground by introducing a rich set of algebraic and combinatorial techniques, particularly focusing on constructions via pseudo-cyclic strongly regular graphs (PC-graphs) and symmetric Hadamard matrices, to produce infinite families of exact values for both diagonal and nearly-diagonal book Ramsey numbers.

Diagonal Book Ramsey Numbers and Strongly Regular Constructions

The pivotal contribution is a new method of constructing lower bounds for $R(B_n, B_n)$ based on the existence of PC-graphs of order $2n-1$. The main result demonstrates that if such a graph exists, then $n$0. This extends the applicability beyond prime power orders (the regime of Paley graphs) to composite orders, leveraging Mathon's product theorem to recursively construct PC-graphs with non-prime-power parameters.

In particular, the authors establish:

• For all $n$1 such that a PC-graph of order $n$2 exists and $n$3 is not the sum of two squares, $n$4. This captures infinite new families of exact diagonal book Ramsey numbers inaccessible by previous Paley-based techniques.
• They concretely identify and classify cases via two algebraic conditions: (i) $n$5 being a prime power, and (ii) $n$6 being a product $n$7 with $n$8 the order of a PC-graph and $n$9 a prime power.

Methodologically, these constructions are grounded in an explicit correspondence between PC-graphs (a class of strongly regular graphs with parameters depending linearly on $R(B_n,B_n)$0) and symmetric conference matrices—objects central to combinatorial design theory. Their analysis provides detailed calculations of common neighbor counts in graph and complement, ensuring the absence of monochromatic $R(B_n,B_n)$1 in either color class.

Nearly-Diagonal Ramsey Numbers and Removal of Parity Constraints

For the off-diagonal regime, the authors investigate $R(B_n,B_n)$2, where only partial results had existed before, typically subject to restrictive congruence conditions on $R(B_n,B_n)$3 or achieved only in sporadic cases via computational or ad hoc combinatorial arguments. Rousseau and Sheehan's previous upper bound of $R(B_n,B_n)$4 applied only when $R(B_n,B_n)$5.

This paper entirely removes this congruence barrier, establishing that:

• $R(B_n,B_n)$6 for all $R(B_n,B_n)$7 except $R(B_n,B_n)$8, regardless of congruence properties of $R(B_n,B_n)$9.

The proof utilizes spectral graph theory to rule out the existence of hypothetical extremal graphs supporting counterexamples, barring the exceptional case $N$0. This is achieved by showing that a candidate strongly regular graph with parameters dictated by the extremal construction does not exist except in this single instance. The case $N$1 is validated as genuine by explicit construction using the complement of the triangular graph $N$2, a well-known strongly regular graph.

Lower Bounds via Hadamard Matrices and Infinite Exact Families

On the lower bound side, the authors construct extremal colorings using symmetric Hadamard matrices of order $N$3 (with all diagonal entries $N$4). The construction yields:

• If such a Hadamard matrix exists, then $N$5.
• For every $N$6 ($N$7), these matrices exist (by Kronecker product), producing another infinite family where $N$8.

The technical crux of these constructions involves careful common neighbor calculations in graphs induced by the Hadamard matrix entries, mirroring the logic for PC-graph constructions but benefiting from the stronger orthogonality conditions of the Hadamard structure.

Numerical Results, Exceptional Cases, and Computational Confirmation

Throughout the exposition, the paper delineates several explicit new values and infinite sequences for both diagonal and nearly-diagonal book Ramsey numbers. Notable special cases and explicit computations are tabulated, and cross-references are made to computational work in the recent literature. For example, the exceptional nonexistence for $N$9 is justified with concrete graph-theoretic examples. Contemporary work using SAT solvers to push lower bounds for small parameters is discussed in relation to these algebraic constructions.

Theoretical and Practical Implications

These advances have multiple implications:

• Extension of exact results: By moving beyond prime-power cases through PC-graphs and new product theorems, the set of known exact book Ramsey numbers is increased dramatically.
• Unified framework: The identification of symmetric conference and Hadamard matrices as universality classes for extremal constructions for (diagonal and off-diagonal) book Ramsey numbers highlights deeper connections between Ramsey theory and design theory.
• Constraints on extremal constructions: The spectral exclusion arguments provide a general template for ruling out potential counterexamples in Ramsey theory by leveraging eigenvalue-based configuration restrictions.
• Template for future work: These techniques could plausibly extend to other graph Ramsey problems where analogous algebraic gadgets can be constructed.

Future Directions

Several open directions are apparent:

• The existence of PC-graphs and symmetric conference/Hadamard matrices for new parameter sets would yield new explicit Ramsey numbers. Classification and construction of such structures is a longstanding active topic in combinatorics.
• These algebraic constructions may admit generalization to higher-order Ramsey numbers involving more complex multipartite or multi-edge graphs.
• Further utilization of computational methods (SAT solvers, automated theorem proving) may work synergistically with algebraic constructions to close gaps or address sparse exceptional cases.

Conclusion

This paper establishes new infinite families of exact Ramsey numbers for book graphs via innovative algebraic constructions utilizing PC-graphs and symmetric Hadamard matrices (2606.07214). By resolving decades-old open questions about both the diagonal and nearly-diagonal cases, removing prior congruence restrictions, and connecting these problems to deep combinatorial/algebraic structures, the work forms a new foundation for further progress in Ramsey theory and extremal combinatorics.