Topological Combinatorics

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this gradually turned into the field of algebraic topology. In 1978 the situation was reversed when methods from algebraic topology were used to solve a problem in combinatorics when László Lovász proved the Kneser conjecture, thus beginning the new study of topological combinatorics. Lovász's proof used the Borsuk-Ulam theorem and this theorem retains a prominent role in this new field. This theorem has many equivalent versions and analogs and has been used in the study of fair division problems. In another application of homological methods to graph theory Lovász proved both the undirected and directed versions of a conjecture of Frank.