A Note on the Turán Number of an Arbitrary Star Forest

The Turán number of a graph G, denoted by ex(n, G), is the maximum number of edges of an n-vertex simple graph having no G as a subgraph. Let Sℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_\ell $$\end{document} denote the star with ℓ+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell +1$$\end{document} vertices, and let ⋃i=1kSℓi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigcup \limits _{i=1}^k S_{\ell _i}$$\end{document} denote the disjoint union of Sℓ1,…,Sℓk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\ell _1},\ldots ,S_{\ell _k}$$\end{document}. For k≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 2$$\end{document} and ℓ1≥⋯≥ℓk≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1\ge \cdots \ge \ell _k\ge 1$$\end{document}, Lidický et al. [On the Turán number of forests, Electron. J. Combin., 20(2)(2013)#P62] proved that ex(n,⋃i=1kSℓi)=max1≤i≤k{⌊(ℓi+2i-3)n-(i-1)(ℓi+i-1)2⌋}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ex(n,\bigcup \limits _{i=1}^k S_{\ell _i})=\max \limits _{1\le i\le k}\{\lfloor \frac{(\ell _i+2i-3)n-(i-1)(\ell _i+i-1)}{2}\rfloor \}$$\end{document} for n sufficiently large. In this paper, we further show that ex(n,⋃i=1kSℓi)=max1≤i≤k(ℓi+2i-3)n-(i-1)(ℓi+i-1)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} ex(n,\bigcup \limits _{i=1}^k S_{\ell _i})=\max \limits _{1\le i\le k}\left\{ \left\lfloor \frac{(\ell _i+2i-3)n-(i-1)(\ell _i+i-1)}{2}\right\rfloor \right\} \end{aligned}$$\end{document}for n≥Nk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge N_k$$\end{document} by giving another proof, where Lj=max1≤i≤j{ℓi+2i-3}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_j=\max \limits _{1\le i \le j} \{\ell _i+2i-3\}$$\end{document} for 2≤j≤k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\le j\le k$$\end{document} and Nk=max2≤j≤k(∑i=1j-1ℓi+1)(∑i=1j2ℓi-ℓj+2j-5)+(j-1)ℓ2-(j-2)ℓj+j2-3j+5Lj-ℓj-2j+5.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_k=\max \limits _{2\le j\le k}\left\{\left\lceil \frac{(\sum \limits _{i=1}^{j-1}\ell _i+1)(\sum \limits _{i=1}^{j} 2\ell _i-\ell _j+2j-5)+(j-1)\ell _2-(j-2)\ell _j+j^2-3j+5}{L_j-\ell _j-2j+5}\right\rceil \right\}.$$\end{document}