|1. The Erdos-Szekeres problem in 3D|
|A set of points in 3-space is said to be in general position if no four of them belong to a plane. It is possible to prove (see, e.g., ) that, for any positive integer n > 3, there is another (smallest) positive integer N = N(n) such that every set of N or more points in general position in 3-space contains a subset of n points which is the vertex set of a convex polytope. It is known that N(4) = 4, N(5) = 6, and N(6) = 9 (see ).|
|Problem. Find the number N(7), or, at least, estimate it.|
 W. Morris, V. Soltan, The Erdos-Szekeres problem, in: J. F. Nash, Jr. and M. Th. Rassias (eds), Open problems in mathematics, pp. 351-375, Springer, 2016.
 T. Bisztriczky, V. Soltan, Some Erdos-Szekeres type results about points in space, Monatsh. Math. 118 (1994), 33-40.
|2. Helly-type problem on common support lines|
|We say that that a line L is a common support for a family F of convex disks in the plane if every disk from F is supported by L. It is possible to prove (see ) the existence of the smallest positive integer k such that any family F of k or more pairwise disjoint convex disks in the plane has a common support line provided every subfamily of three disks from F has a common support line. It is known that 16 < k < 144 (see ).|
|Problem. Find better estimates for the number k.|
 R. Dawson, Helly-type theorems for bodies in the plane with common supports, Geom. Dedicata 45 (1993), 289-299.
 S. Revenko, V. Soltan, Helly-type theorems on transversality for set-system, Studia Sci. Math. Hungar. 32 (1996), 395-406.