Turan number for \(pS_r\)

The Turán number \(ex(m, G)\) of the graph \(G\) is the maximum number of edges of an \(m\)-vertex simple graph having no \(G\) as a subgraph. A \emph{star} \(S_r\) is the complete bipartite graph \(K_{1,r}\) (or a tree with one internal vertex and \(r\) leaves) and \(pS_r\) denotes the disjoint union of \(p\) copies of \(S_r\). A result of Lidický et al. (Electron. J. Combin. \(20(2)(2013) P62\)) implies that \(ex(m,pS_r) = \left\lfloor\frac{(m-p+1)(r-1)}{2}\right\rfloor + (p-1)m – \binom{p}{2}\) for \(m\) sufficiently large. In this paper, we give another proof and show that \(ex(m,pS_r) = \left\lfloor \frac{(m-p+1)(r-1)}{2}\right\rfloor + (p-1)m – \binom{p}{2}\) for all \(r \geq 1\), \(p \geq 1\), and \(m \geq \frac{1}{2}r^2p(p – 1) + p – 2 + \max\{rp, r^2 + 2r\}\).