Extremal Problems for Finite Sets

Extremal Problems for Finite Sets

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One of the great appeals of Extremal Set Theory as a subject is that the statements are easily accessible without a lot of mathematical background, yet the proofs and ideas have applications in a wide range of fields including combinatorics, number theory, and probability theory. Written by two of the leading researchers in the subject, this book is aimed at mathematically mature undergraduates, and highlights the elegance and power of this field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdos matching conjecture. The second half presents some combinatorial structural results and linear algebra methods including the Deza-Erdos-Frankl theorem, application of Rodl's packing theorem, application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdos-Szemeredi sunflower conjecture and capset problem. The book concludes with a collection of challenging open problems.
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Product details

  • Paperback | 224 pages
  • 140 x 216 x 12.7mm | 272.16g
  • Providence, United States
  • English
  • 1470440393
  • 9781470440398
  • 2,095,425

Table of contents

Operations on sets and set systems
Theorems on traces
The Erdos-Ko-Rado theorem via shifting
Katona's circle
The Kurskal-Katona theorem
Kleitman theorem for no $s$ pairwise disjoint sets
The Hilton-Milner theorem
The Erdos matching conjecture
The Ahswede-Khachatrian theorem
Pushing-pulling method
Uniform measure versus product measure
Kleitman's correlation inequality
$r$-cross union families
Random walk method
Exponent of $(10,\{0,1,3,6\})$-system
The Deza-Erdos-Frankl theorem
Furedi's structure theorem
Rodl's packing theorem
Upper bounds using multilinear polynomials
Application to discrete geometry
Upper bounds using inclusion matrices
Some algebraic constructions for $L$-systems
Oddtown and eventown problems
Tensor product method
The ratio bound
Measures of cross independent sets
Application of semidefinite programming
A cross intersection problem with measures
Capsets and sunflowers
Challenging open problems
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About Peter Frankl

Peter Frankl, Renyi Institute, Budapest, Hungary.

Norihide Tokushige, Ryukyu University, Okinawa, Japan.
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