Two-value formula for the Ramsey number of a 4-cycle versus a star

Determine whether, for every integer n≥6, the Ramsey number r(C₄,Bₙ⁽¹⁾) equals n+⌊√(n−1)⌋+γ for some γ∈{1,2}.

Background

The paper reviews that all currently known values of r(C₄,Bₙ⁽¹⁾) for n≥6 have one of two forms: n+⌊√(n−1)⌋+1 or n+⌊√(n−1)⌋+2. It identifies as unresolved whether these are the only possible values for every n≥6. A positive resolution would disprove the subsequent conjecture attributed to Burr, Erdős, Faudree, Rousseau, and Schelp.

References

Note that all known values of the Ramsey number $r(C_4, B_{n}{(1)})$ for $n \geq 6$ satisfy either $r(C_4, B_{n}{(1)})=n+\lfloor\sqrt{n-1}\rfloor+1$ or $r(C_4, B_{n}{(1)})=n+\lfloor\sqrt{n-1}\rfloor+2$. It is an open question, see \, to show whether it is always true that $r(C_4, B_{n}{(1)})=n+\lfloor\sqrt{n-1}\rfloor+\gamma$ for $\gamma \in \{1,2\}$.

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