Extend weakened Gallai–Ramsey numbers for larger books

Determine the values of the weakened Gallai–Ramsey numbers gr^t_2(B_n) for larger values of n by extending the structural and case-based methods developed for books B_5 through B_15, potentially using known two-color Ramsey numbers for books.

Background

The paper determines gr3_2(B_n) for 5≤n≤15 and establishes general lower bounds for grt_2(B_n). Its proofs rely heavily on structural decompositions of Gallai colorings and on known values of two-color Ramsey numbers such as r(B_{n-1},B_n) and r2(B_n). The authors indicate that extending these arguments to larger n is unresolved because the existing proofs do not yet generalize and the necessary Ramsey-number data may be limited.

The unresolved direction is to continue determining weakened Gallai–Ramsey numbers for larger books, exhausting the available two-color Ramsey numbers and extending the techniques used in the paper.

References

We are optimistic that the methods presented in Section~\ref{main} could be continued for larger values of n, repeating the same techniques, with an increasing number of cases. So far, we have been unable to generalize our arguments, which often relied on the known values of $r(B_{n-1},B_n)$ and $r2(B_n)$. One could certainly continue this pursuit, exhausting the known values of $2$-color Ramsey numbers for books.

— Weakened Gallai-Ramsey Numbers for Books  (2608.12673 - Budden et al., 13 Aug 2026) in Section 6, Conclusion