Tur(?)n’s Problem And Ramsey Numbers For Trees With Order N And Maximal Degree N-4

For n>6,letand T_n⁴ =（V,E4）.In this paper,for p>n>10 we obtain explicit formulas for ex（p;T_n⁴）,where ex（p;L）denotes the maximal number of edges in a graph of order p not containing L as a subgraph.Suppose p = k（n-1）+ r,κ ∈ ｛1,2,3,...} and r ∈ {0,1,...,n-2}.If r ∈ ｛3,4,...,n-8},thenwhere b is the least nonnegative residue of n-4 modulo r + 3 and[x]is the greatest integer not exceeding x.For r = n-7 we have ex{p;T_n⁴)=（n-2）p-4(n-4/2.Let r（G₁,G₂）be the Ramsey number of the two graphs G₁ and G₂.In this paper we also obtain some explicit formulas for r（Gm,T_n⁴）,where Gm is connected with order m ≤ n.In particular,if n>m + 2 ≤ 15,then...