There are $p^2$ points and $p^2$ lines. Two lines have no more than one intersection point. Is it possible (for any $p$) to arrange the points so that there are $p$ points on each line?
For example, for $p=2$ it is possible.
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It is clearly impossible when all lines are parallel. But it seems interesting to ask when this is possible. – AdditIdent Nov 09 '18 at 01:15
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It might be useful to count “number of points on each line“ and it adds up to $p^3$. It means “number of lines passing through each point“ also adds up to $p^3$. You could know that there can't be too few intersection points counting multiplicity. This yields necessary condition, but unclear whether sufficient. – AdditIdent Nov 09 '18 at 01:23
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You need indeed not only intersections but high order intersections since $p^3$ overrides $p^2$ when $p$ is large. – AdditIdent Nov 09 '18 at 03:31