Open Problem Garden

It is an easy consequence of Euler's formula that every 3-polytope has a face which is either a triangle, a quadrilateral, or a pentagon. The 120-cell is a 4-polytope in which every 2-face is a pentagon (in fact every 3-face is a regular dodecahedron). Perles and Shephard asked whether there exist higher dimensional polytopes in which all 2-faces have at least 5 vertices. This question was answered in the negative by Kalai [K] who showed that every 5-polytope has a 2-face with at most 4 vertices. So, if we define to be the smallest integer satisfying the above conjecture for , or if none exists, then .

This conjecture is still open for simple polytopes. However, it is known that for every positive integer , there exists an integer so that every simple polytope of dimension either has a 2-dimensional face which is a triangle, or a -dimensional face which is combinatorially isomorphic to a cube. This was proved by Kalai [K] using some earlier results of Nikulin and of Blind and Blind. Actually, something much stronger holds here: simple polytopes of sufficiently high dimension without 2-faces which are triangles must have most -dimensional faces combinatorially isomorphic to the -cube.

The following is an interesting weakening of the above conjecture.

Conjecture   For every positive integer , there exists an integer and a finite list of -dimensional polytopes, so that every polytope of dimension has a -dimensional face which appears in .

Defining to be the smallest integer satisfying this conjecture for , or if none exists, we find that (by the consequence of Euler's formula in the first paragraph). Meisinger, Kleinschmidt, and Kalai [MKK] proved that with the help of FLAGTOOL, a computer program which can compute linear relations for -vectors. This weaker conjecture is known to be true for simple polytopes.