Combinatorial problems on line arrangements

Abstract

In this thesis we discuss two combinatorial problems on arrangements of lines, Karasev's conjecture and Barba's line depth problem. First, we construct counterexamples to the conjecture of Karasev on line arrangements in $R^2$ and show that it holds for lines in convex position. Second, we give a counterexample to Barba's problem on the lower bound of the depth of line arrangements in $R^3$. Here we construct a family of arbitrarily many lines which has depth at most two.