Abstract: In 2005, Goodman and Pollack introduced the concept of
the interval sequence (or double permutation sequence) associated with
a family of pairwise disjoint convex sets in the plane, a combinatorial object
meant to open such families to investigation by combinatorial methods. 
They, Dhandapani, and Holmsen used this concept to address Tverberg's
(1,k)-separation problem:  What is the minimal number of pairwise disjoint
convex sets in the plane required in order to guarantee that one can be
separated from k others by a straight line? (Call this number f_k.) 
Tverberg himself proved that f_k exists for all k and that f_2=5; Hope and
Katchalski proved in 1990 that f_k<=12(k-1). The above authors provided
a new proof that f_2=5 but left the case of general k as an open problem.
We use interval sequences to prove that f_k<=7.2(k-1).