A $$t_k$$ t k Inequality for Arrangements of Pseudolines

Abstract

Consider arrangements of n pseudolines in the real projective plane. Let $$t_k$$tk denote the number of intersection points where exactly k pseudolines are incident. We present a new combinatorial inequality: $$\begin{aligned} t_2+1.5t_3\ge 8+\sum _{k\ge 4}(2k-7.5)t_k, \end{aligned}$$t2+1.5t3?8+?k?4(2k-7.5)tk,which holds if no more than $$n-3$$n-3 pseudolines intersect at one point. It looks similar but is unrelated to the Hirzebruch inequality for arrangements of complex lines in the complex projective plane. Based on this linear inequality, we construct lower bounds for the number of regions via n and the maximal number of (pseudo)lines passing through one point.