Coding and Counting Arrangements of Pseudolines

Arrangements of lines and pseudolines are important and appealing objects for research in discrete and computational geometry. We show that there are at most 2 n 2 simple arrangements of n pseudolines in the plane. This improves on previous work by Knuth who proved an upper bound of 3( n 2) ∼= 2 n in 1992 and the first author who obtained 2 n 2 in 1997. The argument uses surprisingly little geometry. The main ingredient is a lemma that was already central to the argument given by Knuth.