On book-complete graph Ramsey numbers
The book \(B_k\) is the graph consisting of \(m\) triangles sharing one edge. It has been conjectured that the book-complete graph Ramsey number satisfies \(r(B_m,K_n)=(m+1)(n-1)+1\) for a fixed \(n\) and sufficiently large \(m\). The authors prove that for all \(m\geq 1\) and \(n\geq 3\) the inequalities \[ r(B_m,K_n)\leq{mn^2\over\log(n/e)}\quad\text{and}\quad r(K_1+T_m,K_n)\leq{(2m-1)n^2\over\log(n/e)} \] are satisfied. In the last inequality, \(T_m\) is any tree with \(m\) edges. For fixed \(m\), these bounds are asymptotically the best.
scientific article; zbMATH DE number 934823