On Quadrilateral–Book Ramsey Numbers

Let Bn=K1+K1,n be the book with n pages, and write f(n)=R(C4,K1,n) and g(n)=R(C4,Bn). We develop transfer relations that convert information on the star Ramsey numbers f(n) into bounds and exact values for the book Ramsey numbers g(n). Starting from a general inequality of Burr and Erdős, we obtain g(n)≤f(f(n)). We establish the complementary relation g(2n−f(n)+1)≥f(n) and give a sufficient condition for equality in the upper bound. We also derive an edge-counting criterion for graphs witnessing lower bounds for g(n). These results yield several new exact values and improved bounds. In particular, we obtain g(15)=25, g(17)=28, g(19)=30, g(20)=31, g(21)=32, g(n)=n+13 for 22≤n≤30, g(34)=49, g(36)=51, g(41)=56, g(64)=83, g(65)=84, and g(67)=86. Further numerical bounds for n≤72 are collected in a final table.