On the Turan number of forests (Canelled)

A well-known conjecture of Erd\H os and S\'{o}s states that the Tur\'{a}n number for paths is enough for any tree i.e. a graph $G$ on $n$ vertices and more than $\frac{k-2}{2}n$ edges contains any tree on $k$ vertices. A natural extension of the problem is the determination of the Tur\'{a}n number of forests. Erd\H os and Gallai considered the graph $H$ consisting of $k$ independent edges. Brandt generalized Erd\H os and Gallai's result by proving that the Tur\'{a}n number for $k$ independent edges is enough for any forest on $k$ edges without isolated vertices. Bushaw and Kettle found the Tur\'{a}n number and extremal graph for the forest with components, which are paths, of the same order $l>2$. \\ Recently, Lidick\'{y}, Liu and Palmer generalized these results by finding the Tur\'{a}n number and extremal graph for the forests with components of paths with arbitrary length. Also, they investigated Tur\'{a}n number for a star forest $F=\cup_{i=1}^{k}{S_{i}}$ where $d_{i}$ is the maximum degree of $S_{i}$ and $d_{1}\geq \dots \geq d_{k}$, and the Tur\'{a}n number for forests with all components of order 4.\\ In this paper, we determine the Tur\'{a}n number of the forest $F=a\cdot P_{\ell}\cup b\cdot S_{t}$ i.e. $a$ disjoint copies of path $P_{\ell}$ and $b$ disjoint copies of star $S_{t}$. In addition, we obtain the Tur\'{a}n number for forest with all components of order 5.