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Turán's problem and Ramsey numbers for trees

Zhi-Hong Sun;  Lin-Lin Wang;  Yi-Li Wu — 2015

Colloquium Mathematicae

Let T¹ₙ = (V,E₁) and T²ₙ = (V,E₂) 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 ex(p;T¹ₙ) and ex(p;T²ₙ), where ex(p;L) denotes the maximal number of edges in a graph of order p not containing L as a subgraph. Let r(G₁,G₂) be the Ramsey number of the two graphs G₁ and G₂. We also obtain some explicit formulas for r ( T ₘ , T ₙ i ) , where i ∈ 1,2 and Tₘ is a tree on m vertices with Δ(Tₘ) ≤ m - 3.

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