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Pick $n\ge 3$ non-vertical lines $\mathscr{L}:=\{\ell_1,\ldots,\ell_n\}$ in the plane which are pairwise non-parallel, and they are not all concurrent in a single point.

Question. Does there exist a vertical line $L$ which has $n-1$ or $n-2$ intersections with $\mathscr{L}$?

Note: More explicitly, "$L$ has $k$ intersections with $\mathscr{L}$" means that the set $\{L\cap \ell_i: i=1,\ldots,n\}$ has $k$ distinct points.

Partial note 1: By using a dual result due to Ungar here one can see that there are at least $n-1$ vertical lines $L$ which have at most $n-1$ intersections with $\mathscr{L}$

Partial note 2: The Silvester-Gallai problem shows the existence of an ordinary line, that is, given $n$ points in the plane, the exists a connecting line passing through exactly two of those points (this has been improved "optimally" here by Green and Tao). In the setting above, dually, there exists a point which is the intersection of exactly two lines of $\mathscr{L}$.

Partial note 3: This seems to be somehow related to (dual of) Problem 1 at the end of this article, as suggested by domotorp in his answer at this previous question. However, the motivation of the latter problem was a conjecture of Erdos and Purdy, namely, $n$ points in the plane determine at least $\lfloor (n-1)/2\rfloor$ triangles with distinct positive area; finally, this has been proved by Pinchasi in 2008 here.

Pick $n\ge 3$ non-vertical lines $\mathscr{L}:=\{\ell_1,\ldots,\ell_n\}$ in the plane which are pairwise non-parallel, and they are not all concurrent in a single point.

Question. Does there exist a vertical line $L$ which has $n-1$ or $n-2$ intersections with $\mathscr{L}$?

Note: More explicitly, "$L$ has $k$ intersections with $\mathscr{L}$" means that the set $\{L\cap \ell_i: i=1,\ldots,n\}$ has $k$ distinct points.

Partial note 1: By using a dual result due to Ungar here one can see that there are at least $n-1$ vertical lines $L$ which have at most $n-1$ intersections with $\mathscr{L}$

Partial note 2: The Sylvester-Gallai problem shows the existence of an ordinary line, that is, given $n$ points in the plane, the exists a connecting line passing through exactly two of those points (this has been improved "optimally" here by Green and Tao). In the setting above, dually, there exists a point which is the intersection of exactly two lines of $\mathscr{L}$.

Partial note 3: This seems to be somehow related to (dual of) Problem 1 at the end of this article, as suggested by domotorp in his answer at this previous question. However, the motivation of the latter problem was a conjecture of Erdos and Purdy, namely, $n$ points in the plane determine at least $\lfloor (n-1)/2\rfloor$ triangles with distinct positive area; finally, this has been proved by Pinchasi in 2008 here.

Pick $n\ge 3$ non-vertical lines $\mathscr{L}:=\{\ell_1,\ldots,\ell_n\}$ in the plane which are pairwise non-parallel, and they are not all concurrent in a single point.

Question. Does there exist a vertical line $L$ which has $n-1$ or $n-2$ intersections with $\mathscr{L}$?

Note: More explicitly, "$L$ has $k$ intersections with $\mathscr{L}$" means that the set $\{L\cap \ell_i: i=1,\ldots,n\}$ has $k$ distinct points.

Partial note 1: By using a dual result due to Ungar here one can see that there are at least $n-1$ vertical lines $L$ which have at most $n-1$ intersections with $\mathscr{L}$

Partial note 2: The Silvester-Gallai problem shows the existence of an ordinary line, that is, given $n$ points in the plane, the exists a connecting line passing through exactly two of those points (this has been improved here by Green and Tao, solving a conjecture of Erdos). In the setting above, dually, there exists a point which is the intersection of exactly two lines of $\mathscr{L}$.

Partial note 3: This seems to be somehow related to (dual of) Problem 1 at the end of this article, as suggested by domotorp in his answer at this previous question. However, the motivation of the latter problem was a conjecture of Erdos and Purdy, namely, $n$ points in the plane determine at least $\lfloor (n-1)/2\rfloor$ triangles with distinct positive area; finally, this has been proved by Pinchasi in 2008 here.

Pick $n\ge 3$ non-vertical lines $\mathscr{L}:=\{\ell_1,\ldots,\ell_n\}$ in the plane which are pairwise non-parallel, and they are not all concurrent in a single point.

Question. Does there exist a vertical line $L$ which has $n-1$ or $n-2$ intersections with $\mathscr{L}$?

Note: More explicitly, "$L$ has $k$ intersections with $\mathscr{L}$" means that the set $\{L\cap \ell_i: i=1,\ldots,n\}$ has $k$ distinct points.

Partial note 1: By using a dual result due to Ungar here one can see that there are at least $n-1$ vertical lines $L$ which have at most $n-1$ intersections with $\mathscr{L}$

Partial note 2: The Silvester-Gallai problem shows the existence of an ordinary line, that is, given $n$ points in the plane, the exists a connecting line passing through exactly two of those points (this has been improved "optimally" here by Green and Tao). In the setting above, dually, there exists a point which is the intersection of exactly two lines of $\mathscr{L}$.

Partial note 3: This seems to be somehow related to (dual of) Problem 1 at the end of this article, as suggested by domotorp in his answer at this previous question. However, the motivation of the latter problem was a conjecture of Erdos and Purdy, namely, $n$ points in the plane determine at least $\lfloor (n-1)/2\rfloor$ triangles with distinct positive area; finally, this has been proved by Pinchasi in 2008 here.