Before revision:  There are several variations on this notion in the literature, one of which requires that no three pseudocircles have a point in common. Working under this definition, Ortner proved that an arrangement of pseudocircles is embeddable into the sphere if and only if all of its subarrangements of size at most 4 are embeddable into the sphere .
After revision:  Ortner proved that an arrangement of pseudocircles is embeddable into the sphere if and only if all of its subarrangements of size at most 4 are embeddable into the sphere .