Question 137
We call a set of lines in the plane nice if every line in the set intersects an odd number of the other lines in the set.
Determine the smallest integer $k \geq 0$ with the following property: for any 2018 distinct lines $\ell_{1}, \ell_{2}, \ldots, \ell_{2018}$ in the plane there exist lines $\ell_{2018+1}, \ell_{2018+2}, \ldots, \ell_{2018+k}$ such that the lines $\ell_{1}, \ell_{2}, \ldots, \ell_{2018+k}$ are all distinct and form a nice set.
Determine the smallest integer $k \geq 0$ with the following property: for any 2018 distinct lines $\ell_{1}, \ell_{2}, \ldots, \ell_{2018}$ in the plane there exist lines $\ell_{2018+1}, \ell_{2018+2}, \ldots, \ell_{2018+k}$ such that the lines $\ell_{1}, \ell_{2}, \ldots, \ell_{2018+k}$ are all distinct and form a nice set.