Random Polynomial Graphs for Random Turán Problems

Bukh and Conlon used random polynomial graphs to give effective lower bounds on $\mathrm{e}\mathrm{x}(n,{\mathcal{T}}^{\mathrm{\ell }})$, where ${\mathcal{T}}^{\mathrm{\ell }}$ is the $\mathrm{\ell }$th power of a balanced rooted tree $T$. We extend their result to give effective lower bounds on $\mathrm{e}\mathrm{x}(G_{n,p},{\mathcal{T}}^{\mathrm{\ell }})$, which is the maximum number of edges in a ${\mathcal{T}}^{\mathrm{\ell }}$-free subgraph of the random graph $G_{n,p}$. Analogous bounds for generalized Turán numbers in random graphs are also proven.