Forcing subarrangements in complete arrangements of pseudocircles

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 α-subarrangement, a certain arrangement of three pseudocircles. Similarly, 4n−5 vertices of weight 0 force an α-subarrangement (of four pseudocircles). These results on the one hand give improved bounds on the number of vertices of weight ≤ k for complete, α-free and complete, α-free arrangements. On the other hand, interpreting αand α-arrangements as complete graphs with three and four vertices, respectively, the bounds correspond to known results in extremal graph theory.