Line covers after random deletions

Fix a positive integer $D$, reals $\delta,\rho>0$ and $\eta>0$. There is a threshold $Q$ such that the following holds for every finite field $F$ with $q=|F|\ge Q$ all of whose proper subfields $M$ satisfy $|M|^3\le q$, and for every deletion process on $F^2$ with parameters $D,\delta$.

Such a process consists of at most $Dq^D$ shapes in $F^2$, each with at most $Dq$ points, and each meeting any different affine line in at most $D$ points, together with at most $Dq$ independent trials, each selecting at most $D$ shapes. Each trial deletes a given point with probability at most $D/q$; each point survives all trials with the same probability, which is at least $\delta$; and for each affine line $\ell$, the probabilities that the trials list exactly the shape $\ell$ sum to at most $D/q$. Let $X$ be the surviving set, and let $X^\ell$ be the surviving set when exact occurrences of the shape $\ell$ are ignored.

Then with probability at least $1-\eta$, both of the following hold:

and every set $L$ of at most $q$ affine lines leaving at most $2q$ points of $X$ uncovered contains at least $(1-\rho)q$ lines through a common projective point.

This is Lemma 3.2 of the paper, in a form uniform over the field and the process. It is the source of the rigidity of small covers: the ordinary edges of the hypergraph force any cover of size at most $q$ to contain almost a full pencil.