Problems

Prove Sperner’s lemma in dimension \(1\), namely on a line.
The simplex in this case is just a segment, the triangulation is subdivision of the segment into multiple small segments, and the conditions of a Sperner’s coloring are the following:

• There are only two colors;

• The opposite ends of the main segment are colored differently;

Then one needs to prove that there exists a small segment with two ends colored in different colors. In particular there is an odd number of such small segments.

Draw a Sperner’s coloring for the following \(3\)-dimensional simplex. The blue segments are visible, the grey ones are inside the tetrahedron. The point \(F\) is on the face \(ABC\), point \(E\) is on the face \(BCD\), point \(G\) is on the face \(ACD\) and the point \(H\) is on the face \(ABD\).

Draw Sperner’s coloring for the following triangulation. Try to avoid rainbow triangles at all costs.