# Peter Keevash

Peter Keevash (born November 30, 1978 in Brighton ) is a British mathematician. Keevash specializes in combinatorics .

## Life

Keevash grew up in Leeds , took part in the International Mathematical Olympiad in 1995 (bronze medal) and studied mathematics at Cambridge University (Trinity College) from 1995 with a bachelor's degree in 1998. He was trained in Princeton in 2004 by Benny Sudakov on The Role of Approximate Structure in Extremal Combinatorics and was a post-doctoral student at Caltech . He was a lecturer and then professor at Queen Mary College, University of London, and has been a professor at Oxford University since 2013 . He is a Tutorial Fellow at Mansfield College, Oxford.

## plant

He deals with extremal combinatorics, graph theory , hypergraphs , algebraic and probabilistic methods in combinatorics, random structures in combinatorics, combinatorial optimization and combinatorial number theory.

In 2014 he proved an important problem of combinatorics that had been open for a long time, the question of the existence of combinatorial designs ( block plans ) for any values ​​of the parameters, whereby these must meet certain natural divisibility conditions. He proved that for all k, t and such designs exist for all numbers v that meet the divisibility conditions mentioned, apart from a finite number of exceptions. For t = 2 Richard M. Wilson had already proved the existence of sufficiently large feasible v from 1972 to 1975. In 2015 Keevash also found an approximate estimate for the number of designs with certain parameters, also a long-standing problem. He proved and generalized a conjecture by Richard M. Wilson from 1974, who formulated it for Steiner-Triple systems. Keevash used the Randomized Algebraic Construction method he had developed . Examples of designs and Steiner systems with t greater than 2 were only incompletely known, and Keevash's theorem proved their existence even for any t. ${\ displaystyle S _ {\ lambda} (t, k, v)}$${\ displaystyle \ lambda}$

The question of the existence of designs with certain parameters goes back to Julius Plücker (1835) and Thomas Kirkman (1847) and Jakob Steiner (1853).

In 2013 he and Tom Bohman proved the best known lower bound for the Ramsey number R (k, 3) in Ramsey theory .

In 2009 he received the European Prize in Combinatorics . For 2015 he was awarded the Whitehead Prize of the London Mathematical Society.

## Fonts (selection)

• with T. Bohman: The early evolution of the H-free process , Inventiones Mathematicae 181 (2010), 291-336.
• with R. Mycroft: A geometric theory of hyper graph matching , Mem.AMS 233 (2014)
• The existence of designs , arxiv.org/abs/1401.3665

## literature

• WT Gowers: Probabilistic combinatorics and the recent work of Peter Keevash, Bulletin AMS 2016, Online

2. The blocks of such a block plan are k-element subsets of a set P with v elements. It is required that every subset of P with t elements is contained in exactly blocks. In this case one speaks of Steiner systems.${\ displaystyle \ lambda}$${\ displaystyle \ lambda = 1}$