Syllabus
Simplicial complexes, connectedness of a space.
Borsuk-Elam theorem, equivalent versions.
Ham-sandwich theorem, Necklace theorem.
Theorems on non-embeddability and colorings (chromatic number of Kneser graphs, Radon theorem).
Additional (possible) topics: homology, degree of a map, colorful Tverberg theorem, Z_2 index.
Annotation
One of the important proof techniques in discrete mathematics is the application of theorems from algebraic topology.
The course covers the necessary topological preliminaries and establishes several combinatorial and geometric results by topological methods, mainly using the Borsuk-Ulam theorem.