Forcing subarrangements in complete arrangements of pseudocircles

Abstract

In arrangements of pseudocircles (i.e., Jordan curves) the weight of a vertex (i.e., an intersection point) is the number of pseudocircles that contain the vertex in its interior. We show that in complete arrangements (in which each two pseudocircles intersect) $2n-1$ vertices of weight 0 force an $\alpha$ -subarrangement, a certain arrangement of three pseudocircles. Similarly, $4n-5$ vertices of weight 0 force an $\alpha^4$-subarrangement (of four pseudocircles). These results on the one hand give improved bounds on the number of vertices of weight $k$ for complete, $\alpha$-free and complete, $\alpha^4$-free arrangements. On the other hand, interpreting $\alpha$- and $\alpha^4$-arrangements as complete graphs with three and four vertices, respectively, the bounds correspond to known results in extremal graph theory.