Extremal Problems for Finite Sets

Description

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 Erd s matching conjecture. The second half presents some combinatorial structural results and linear algebra methods including the Deza Erd s Frankl theorem, application of Rödl's packing theorem, application of semidefinite programming, and very recent progress (obtained in 2016) on the Erd s Szemerédi sunflower conjecture and capset problem. The book concludes with a collection of challenging open problems. Cover......Page 1 Title page......Page 4 Contents......Page 6 Notation......Page 8 Chapter 1. Introduction......Page 10 Chapter 2. Operations on sets and set systems......Page 14 Chapter 3. Theorems on traces......Page 22 Chapter 4. The Erdős–Ko–Rado Theorem via shifting......Page 28 Chapter 5. Katona’s circle......Page 32 Chapter 6. The Kruskal–Katona Theorem......Page 40 Chapter 7. Kleitman Theorem for no �� pairwise disjoint sets......Page 46 Chapter 8. The Hilton–Milner Theorem......Page 52 Chapter 9. The Erdős matching conjecture......Page 56 Chapter 10. The Ahlswede–Khachatrian Theorem......Page 62 Chapter 11. Pushing-pulling method......Page 70 Chapter 12. Uniform measure versus product measure......Page 78 Chapter 13. Kleitman’s correlation inequality......Page 86 Chapter 14. ��-Cross union families......Page 92 Chapter 15. Random walk method......Page 96 Chapter 16. ��-systems......Page 104 Chapter 17. Exponent of a (10,{0,1,3,6})-system......Page 112 Chapter 18. The Deza–Erdős–Frankl Theorem......Page 118 Chapter 19. Füredi’s structure theorem......Page 124 Chapter 20. Rödl’s packing theorem......Page 130 Chapter 21. Upper bounds using multilinear polynomials......Page 136 Chapter 22. Application to discrete geometry......Page 146 Chapter 23. Upper bounds using inclusion matrices......Page 150 Chapter 24. Some algebraic constructions for ��-systems......Page 158 Chapter 25. Oddtown and eventown problems......Page 164 Chapter 26. Tensor product method......Page 170 Chapter 27. The ratio bound......Page 184 Chapter 28. Measures of cross independent sets......Page 190 Chapter 29. Application of semidefinite programming......Page 198 Chapter 30. A cross intersection problem with measures......Page 204 Chapter 31. Capsets and sunflowers......Page 210 Chapter 32. Challenging open problems......Page 220 Bibliography......Page 226 Index......Page 232 Back Cover......Page 234