All missing Ramsey numbers for trees versus the four-page book

Abstract

For the Ramsey number r(Tn, Bm), where Tn denotes a tree of order n and Bm denotes the m-page book K2 + bar(K)m, it is known that r(Tn, Bm)=2n − 1 if n ≥ 3m − 3. In case of n < 3m − 3, r(Tn, Bm) has not been completely evaluated except for m ≤ 3. Here we determine the missing values of r(Tn, B4). Our results close one gap in the table of the Ramsey numbers r(Tn, G) for all trees Tn and all connected graphs G of order six.

Keywords

• ramsey number, tree, book, small graph