Graduate Research Projects in Disrete Geometry

 
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., [1]) 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 [2]).
Problem. Find the number N(7), or, at least, estimate it.
[1] 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.
[2] 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 disjoint family F of convex disks in the plane if every disk from F is supported by L. It is possible to prove (see [1]) the existence of a smallest positive integer k such that any disjoint family F of k or more 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 [2]).
Problem. Find better estimates for the number k.
[1] R. Dawson, Helly-type theorems for bodies in the plane with common supports, Geom. Dedicata 45 (1993), 289-299.
[2] S. Revenko, V. Soltan, Helly-type theorems on transversality for set-system, Studia Sci. Math. Hungar. 32 (1996), 395-406.