Ramsey Numbers for Trees II

Let r ( G 1 , G 2 ) be the Ramsey number of the two graphs G 1 and G 2 . For n 1 ≽ n 2 ≽ 1 let S ( n 1 , n 2 ) be the double star given by V ( S ( n 1 , n 2 ) ) = { v 0 , v 1 , … , v n 1 , w 0 , w 1 , … , w n 2 } and E ( S ( n 1 , n 2 ) ) = { v 0 v 1 , … , v 0 v n 1 , v 0 w 0 , w 0 w 1 , … , w 0 w n 2 } . We determine r ( K 1, m −1 , S ( n 1 , n 2 )) under certain conditions. For n ≽ 6 let T n 3 − S ( n − 5, 3), T n ″ − ( V, E 2 ) and T n ‴ = ( V, E 3 ), where V = { v 0 , v 1 , …, v n −1 }, E 2 = { v 0 v 1 , …, v 0 v n −4 , v 1 v n −3 , v 1 v n −2 , v 2 v n −1 } and E 3 = { v 0 v 1 , …, v 0 v n −4 , v 1 v n −3 , v 2 v n −2 , v 3 v n −1 }. We also obtain explicit formulas for r ( K 1, m −1 , T n ), r ( T m ′ , T n ) ( n ≽ m + 3), r ( T n , T n ), r ( T n ′ , T n ) and r ( P n , T n ), where T n ∈ { T n ″ , T n ‴ , T n 3 }, P n is the path on n vertices and T n ′ is the unique tree with n vertices and maximal degree n − 2.