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Beyond boundaries and extreme values

EU-funded research has extended the extremal approach to the solution of problems, including some that were regarded as algorithmic and broadened the choice of fundamental bases for solving practical problems.
Beyond boundaries and extreme values
The term extremal describes the nature of problems that combinatorial mathematics addresses. Specifically, extremal combinatorics centres on a search for the maximum or minimum number of a collection of finite objects – numbers, graphs and vectors – satisfying certain restrictions.

Besides classical tools, new tools have shown their power in solving such problems. Moreover, they have found striking applications in other areas of mathematics as well as the theory of computing. The PECTA (Extremal problems in combinatorics and their applications) project focused on still open problems.

Researchers investigated different aspects of the classical removal lemma from graph theory. This roughly says that a particular graph can be removed from a larger graph containing a small number of copies if some edges are deleted. Green's variant of the lemma proved to have applications in theoretical computer science.

In addition, the PECTA team established conditions under which deterministic structures behave like random ones. This concept turned out to be extremely useful for tackling a variety of problems related to graphs and hypergraphs as well as to extend further the theory of quasi-randomness.

Techniques developed as a result of the project have applications in several areas of mathematics and are described in publications in some of the top mathematical journals. Moreover, many of the problems addressed arise by questions in computer science and therefore have practical applications.

Related information

Keywords

Extremal, combinatorial mathematics, combinatorics, PECTA, graph theory
Record Number: 191163 / Last updated on: 2017-02-21
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