25 years proof of the Kneser conjecture The advent of topological combinatorics

With this article I want to take the opportunity to sketch the development of topological proofs in discrete mathematics beginning with the proof of the Kneser conjecture about 25 years ago, which eventually led to a new discipline: topological combinatorics. In the beginning of the twentieth century the discipline of combinatorial topology already made use of combinatorial concepts in topology, finally leading to the emergence of algebraic topology. Meanwhile, discrete mathematics did not make much use of techniques from (algebraic) topology until the seminal proof of the Kneser conjecture. This situation was going to change in an unexpected and fascinating way. The essence of topological combinatorics can be characterized by a scheme that many proofs in this field pursue. If we want to solve a combinatorial problem by topological means we carry out the following steps.