Burr–Erdős–Faudree–Rousseau–Schelp conjecture

Prove or disprove that, for every constant c>0, the Ramsey number r(C₄,Bₙ⁽¹⁾) is less than n+√n−c for infinitely many integers n.

Background

The conjecture concerns the asymptotic behavior of the Ramsey number of a 4-cycle versus a star. The paper notes that a positive answer to the preceding two-value question would disprove this conjecture, and that Erdős offered a $100 prize for its proof or disproof.

References

If one can answer this question in a positive way, then it will disprove the following conjecture due to Burr, Erd\H{o}s, Faudree, Rousseau, and Schelp . \begin{conjecture} For any constant $c>0$, we have $r(C_4, B_{n}{(1)})< n+\sqrt{n}-c$ for infinitely many $n$. \end{conjecture}

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