On Book-Complete Graph Ramsey Numbers

It is shown that a graph of orderNand average degreedthat does not contain the bookBm=K1+K1, mas a subgraph has independence number at leastNf(d), wheref(x)∼(log x/x) (x→∞). From this result we find that the book-complete graph Ramsey number satisfiesr(Bm, Kn)⩽mn2/log(n/e). It is also shown that for every treeTmwithmedges,r(K1+Tm, Kn)⩽(2m−1) n2/log(n/e).