Maximum Induced Trees in Sparse Random Graphs

We prove that for any \(\varepsilon > 0\) and \({{n}^{{ - \frac{{e - 2}}{{3e - 2}} + \varepsilon }}} \leqslant p = o(1)\) the maximum size of an induced subtree of the binomial random graph \(G(n,p)\) is concentrated asymptotically almost surely at two consecutive points.