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Non-constructive proofs in Combinatorics Department of Mathematics

Summary: Non-constructive proofs in Combinatorics
Noga Alon
Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
and IBM Almaden Research Center
San Jose, CA 95120, USA
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 applica-
tions of Topological, Algebraic and Probabilistic methods in Combinatorics, and discuss the related
algorithmic problems.
1 Topological methods
The application of topological methods in the study of combinatorial objects like partially ordered
sets, graphs, hypergraphs and their coloring have become in the last ten years part of the math-
ematical machinery commonly used in combinatorics. Many interesting examples appear in [12].


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics