Question. Consider $n \geq 5$ lines in a general position (i.e. no two lines are parallel and no triple intersections are allowed) in $\mathbb{R}^2$. Let $T(n)$ denote the maximal number of empty triangles (here empty triangle means that it does not contain other triangle). What would be best upper and lower bounds for $T(n)$? I know $(n-2) \leq T(n)$ holds, but I am hoping for a better lower bound. Is it true that $T(6) = 6$n \leq T(n)$? Also, is it possible to compute $T(n)$ it for small $n$ (where small means $6 \leq n \leq 10$) 10$)? I think $T(6) = 6$, but I am not able to show $6$ is an upper bound as well.
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$n$ lines in a general position and the number of empty trianglesQuestion. Consider $n \geq 5$ lines in a general position (i.e. no two lines are parallel and no triple intersections are allowed) in $\mathbb{R}^2$. Let $T(n)$ denote the maximal number of empty triangles (here empty triangle means that it does not contain other triangle). What would be best upper and lower bounds for $T(n)$? I know $(n-2) \leq T(n)$ holds, but I am hoping for a better lower bound. Is it true that $T(6) = 6$? Also, is it possible to compute $T(n)$ it for small $n$ (where small means $6 \leq n \leq 10$)
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