Upper bounds for the number of substructures in finite geometries from the container method

We use techniques from algebraic and extremal combinatorics to derive upper bounds on the number of independent sets in several (hyper)graphs arising from finite geometry. In this way, we obtain asymptotically sharp upper bounds for partial ovoids and EKR-sets of flags in polar spaces, line spreads in $\mathrm{PG}(2r-1,q)$ and plane spreads in $\mathrm{PG}(5,q)$, and caps in $\mathrm{PG}(3,q)$. The latter result extends work due to Roche-Newton and Warren and Bhowmick and Roche-Newton. Finally, we investigate caps in $p$-random subsets of $\mathrm{PG}(r,q)$, which parallels recent work for arcs in projective planes by Bhowmick and Roche-Newton, and by Roche-Newton and Warren, and arcs in projective spaces by Chen, Liu, Nie and Zeng.