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Let $p$ be a fixed prime, and suppose that $S$ is a subset of the affine plane $\mathbb F_p^2$. If $|S|\le p+1$, then by the pigeonhole principle, through any given point $s\in S$ there is a line $L=L(s)$ avoiding all other points of $S$; that is, $L(s)\cap S=\{s\}$. Are any stability results known? I am seeking something of the following sort: if $|S|<(1+\delta)p$, then through almost any point $s\in S$ there is a line avoiding all other points of $S$ $-$ unless $S$ possesses some rigid structure.

Any references will be appreciated.

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    $\begingroup$ This might be useful to you: link.springer.com/article/10.1007/s10623-011-9495-z. The stability they are talking about is different from what you are asking, but perhaps the techniques can be adapted to what you want to show. $\endgroup$
    – Anurag
    Commented Sep 25, 2017 at 11:17

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