Ramsey numbers of the quadrilateral versus books

A book B-n is a graph which consists of n triangles sharing a common edge. In this paper, we study Ramsey numbers of quadrilateral versus books. Previous results give the exact value of r (C-4, B-n) for 1 <= n <= 14. We aim to determine the exact value of r (C-4, B-n) for infinitely many n. To achieve this, we first prove that r (C-4, B-(m -1)(2) +(t-2)) <= m(2) + t 2 for m >= 4 and 0 <= t <= m - 1. This improves upon a result by Faudree, Rousseau, and Sheehan which states that r (C-4, B-n) <= g (g (n)), where g (n) = n + left floor root n - 1 right floor + 2. Combining the new upper bound and constructions of C4-free graphs, we are able to determine the exact value of r (C4, Bn) for infinitely many n. As a special case, we show r (C-4, B-q(2) -q-2) = q(2) + q - 1 for all prime powers q >= 4.