$k$-arrangements of pseudolines and pseudocircles

Source abstract

A $k$-arrangement of pseudolines is a set of bi-infinite curves in the plane such that any two of them intersect each other in exactly $k$ points, at which they cross, and it is simple if no three curves meet at a common point. Cyclic arrangements are the only simple $1$-arrangements of pseudolines that are unavoidable, in the Ramsey spirit: for each fixed $m\ge 1$, every sufficiently large simple $1$-arrangement of pseudolines has a cyclic subarrangement of size $m$. We show that, for every $m\ge 3$, the number of unavoidable simple $k$-arrangements of pseudolines of size $m$ grows exponentially with $k$, independently of $m$. For even $k$, we prove an analogous result for $k$-arrangements of pseudocircles.