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Turan-type problems for graphs and hypergraphs

Project description

Turan-type problems under study

A central question in graph theory is determining how many edges a graph has if it does not contain a specified configuration as a subgraph. Mantel’s theorem from 1908 gives the maximum number of edges a graph can have on a given number of vertices if it does not contain a triangle. Extremal problems of this type are known as Turan-type problems. Funded by the Marie Skłodowska-Curie Actions programme, the TurantypeProblems project will study several Turan-type problems concerning graphs and hypergraphs as well as related extremal problems on rainbow structures.

Objective

In this project, we propose to study several Turan-type extremal problems for graphs and hypergraphs as well as related extremal problems on rainbow structures. In Turan-type extremal problems, roughly speaking, we want to determine how dense a configuration (graph, hypergraph, set system, etc.) can be when certain sub-configurations are forbidden. One of the earliest results in the area is Mantel's Theorem from 1908, which says that the largest triangle-free graph on given number of vertices is the balanced complete bipartite graph. This was generalized by Turan in 1941 to complete graphs on any number of vertices. The Turan-number of a graph G is the largest number of edges that a graph on given number of vertices can have without containing G.

The experienced researcher in recent years has made several contributions to these problems. On Turan-type problems for bipartite graphs, in joint work with Jiang and Ma, the she made a first non-trivial progress in a while to a conjecture of Erdos and Simonovits regarding the so-called Turan exponents of single bipartite graphs. On hypergraph Turan-type problems, in joint work with Sergey Norin, she has developed a highly effective variant of the classical stability method that allowed them to solve several open conjectures in the field. The TC-supervisor, Dhruv Mubayi, is a world leading expert on Turan-type extremal problems. Working on problems posed in this proposal will allow the experienced researcher to further develop these promising tools with a leading expert.

The second set of problems concern rainbow structures in edge-coloured graphs and hypergraphs. The experienced researcher, in joint works with Peter Keevash, and also with others had made several contributions to this area in the last two years. Here there is also a common theme of a series of work done by the MC-supervisor Julia Boettcher, who is a leading expert on the study of spanning structures in graphs and hypergraphs.

Coordinator

LONDON SCHOOL OF ECONOMICS AND POLITICAL SCIENCE
Net EU contribution
€ 189 099,84
Address
Houghton Street 1
WC2A 2AE London
United Kingdom

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Region
London Inner London — West Westminster
Activity type
Higher or Secondary Education Establishments
Links
Total cost
€ 189 099,84

Partners (1)