Existence of Ramsey graphs for the q-parameterized book graph

Determine whether the set of (C₄,B_{q²−q+1}⁽²⁾)-Ramsey graphs on q²+q+3 vertices is nonempty for every prime power q≥5.

Background

For prime-power q, the cited earlier work gives bounds for r(C₄,B_{q²−q+1}⁽²⁾). If the set 𝒢(q) of Ramsey graphs on q²+q+3 vertices is nonempty, then the upper bound is attained exactly. The paper records that 𝒢(2) is empty, 𝒢(3) is nonempty, and 𝒢(4) is empty; it explicitly states that the status for all prime powers q≥5 remains unresolved.

References

Faudree, Rousseau and Sheehan \ proved that $\mathcal{G}(3) \not =\emptyset$ and asked to determine whether $\mathcal{G}(q)\neq\emptyset$ for each prime power $q>3$. The answer to this problem is negative for $q=4$ as $r(C_4,B_{13}{(2)})=22$, see \, which implies $\mathcal{G}(4)=\emptyset$. The problem is wide open for $q\ge5$.

— The Ramsey number of the 4-cycle versus a book graph  (2506.10477 - Dou et al., 12 Jun 2025) in Section 1, Introduction