TURAN'S PROBLEM AND RAMSEY NUMBERS FOR TREES

Let T-n(1) = (V, E-1) and T-n(2) = (V, E-2) be the trees on n vertices with V = {v(0), v (1), ...,v(n-1)}, E-1 = {v(0)v(1),...,v(0)v(n-3), v(n-4)v(n-2), v(n-3)v(n-1)} and E-2 = {v(0)v(1),...,v(0)v(n-3),v(n-3)v(n-2),v(n-3)v(n-1)}. For p >= n >= 5 we obtain explicit formulas for e x (p; T-n(1)) and ex (p; T-n(2)), where ex (p; L) denotes the maximal number of edges in a graph of order p not containing L as a subgraph. Let r (G(1); G(2)) be the Ramsey number of the two graphs G(1) and G(2). We also obtain some explicit formulas for r (T-m, T-n(i)), where i is an element of {1; 2} and T-m is a tree on m vertices with Delta(Tm) <= m - 3