Non-constructive proofs in Combinatorics

One of the main reasons for the fast development of Combinatorics during the recent years is certainly the widely used application of combinatorial methods in the study and the development of efficient algorithms. It is therefore somewhat surprising that many results proved by applying some of the modern combinatorial techniques, including Topological methods, Algebraic methods, and Probabilistic methods, merely supply existence proofs and do not yield efficient (deterministic or randomized) algorithms for the corresponding problems. We describe some representing non-constructive proofs of this type, demonstrating the applications of Topological, Algebraic and Probabilistic methods in Combinatorics, and discuss the related algorithmic problems.

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