Extremal Problems for Finite Sets (Student Mathematical Library)

234 Pages·2018·1.075 MB·English

by  Peter Frankl, Norihide Tokushige

STUDENT MATHEMATICAL LIBRARY
Volume 86
Extremal
Problems for
Finite Sets
Peter Frankl
Norihide Tokushige
Extremal
Problems for
Finite Sets

STUDENT MATHEMATICAL LIBRARY
Volume 86
Extremal
Problems for
Finite Sets
Peter Frankl
Norihide Tokushige
Editorial Board
SatyanL. Devadoss John Stillwell (Chair)
Erica Flapan Serge Tabachnikov
2010 Mathematics Subject Classification. Primary 05-01, 05D05.
For additional informationand updates on this book, visit
www.ams.org/bookpages/stml-86
Library of Congress Cataloging-in-Publication Data
Names: Frankl,P.(Peter),1953-author. |Tokushige,Norihide,1963-author.
Title: Extremalproblemsforfinitesets/PeterFrankl,NorihideTokushige.
Description: Providence, Rhode Island: AmericanMathematicalSociety, [2018]
| Series: Student mathematicallibrary ; volume 86 | Includes bibliographical
referencesandindex.
Identifiers: LCCN2017061643|ISBN9781470440398(alk. paper)
Subjects: LCSH: Extremal problems (Mathematics)| Set theory. | AMS: Com-
binatorics – Instructional exposition (textbooks, tutorial papers, etc.). msc |
Combinatorics–Extremalcombinatorics–Extremalsettheory. msc
Classification: LCCQA295.F8552018|DDC511/.6–dc23
LCrecordavailableathttps://lccn.loc.gov/2017061643
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Contents
Notation vii
Chapter 1. Introduction 1
Chapter 2. Operations on sets and set systems 5
Chapter 3. Theorems on traces 13
Chapter 4. The Erd˝os–Ko–Rado Theorem via shifting 19
Chapter 5. Katona’s circle 23
Chapter 6. The Kruskal–Katona Theorem 31
Chapter 7. Kleitman Theorem for no s pairwise disjoint sets 37
Chapter 8. The Hilton–Milner Theorem 43
Chapter 9. The Erd˝os matching conjecture 47
Chapter 10. The Ahlswede–Khachatrian Theorem 53
Chapter 11. Pushing-pulling method 61
Chapter 12. Uniform measure versus product measure 69
Chapter 13. Kleitman’s correlation inequality 77
v
vi Contents
Chapter 14. r-Cross union families 83
Chapter 15. Random walk method 87
Chapter 16. L-systems 95
Chapter 17. Exponent of a (10,{0,1,3,6})-system 103
Chapter 18. The Deza–Erdo˝s–Frankl Theorem 109
Chapter 19. Fu¨redi’s structure theorem 115
Chapter 20. R¨odl’s packing theorem 121
Chapter 21. Upper bounds using multilinear polynomials 127
Chapter 22. Application to discrete geometry 137
Chapter 23. Upper bounds using inclusion matrices 141
Chapter 24. Some algebraic constructions for L-systems 149
Chapter 25. Oddtown and eventown problems 155
Chapter 26. Tensor product method 161
Chapter 27. The ratio bound 175
Chapter 28. Measures of cross independent sets 181
Chapter 29. Application of semidefinite programming 189
Chapter 30. A cross intersection problem with measures 195
Chapter 31. Capsets and sunflowers 201
Chapter 32. Challenging open problems 211
Bibliography 217
Index 223
Notation
• Let R be the set of real numbers.
• Let Q be the set of rational numbers.
• Let Z be the set of integers.
• Let Z ={1,2,3,...} be the set of positive integers.
>0
• Let N={0,1,2,...} be the set of nonnegative integers.
• Let Z/mZ be the ring of congruence classes modulo m.
• Let F denote the q-element field.
q
• Let Rm×n denote the set of m×n real matrices.
• Let F [x] denote the set of polynomials in variable x with
q
coefficients in F .
q
• For a positive integer n, let [n]={1,2,...,n}.
• For a finite set A, the size (cardinality) of A is denoted by
|A| or #A.
• For functions f,g :R→R we write
– f(x)=o(g(x)) if limx→∞|f(x)/g(x)|=0,
– f(x)=O(g(x))if|f(x)|<Cg(x)forallx>x ,where
0
x and C are some fixed constants,
0
– f(x)=Ω(g(x)) if g(x)=O(f(x)),
– f(x)=Θ(g(x))iff(x)=O(g(x))andf(x)=Ω(g(x)),
– f(n)(cid:4)g(n) if limn→∞f(n)/g(n)=0,
vii
viii Notation
– f(n)∼g(n) if limn→∞f(n)/g(n)=1.
• (Kronecker’s delta) δ =1 if i=j and δ =0 if i(cid:6)=j.
ij ij
• For integers a and b we write a|b to mean that a divides b.
• (Disjoint union) We writeA(cid:7)B forA∪B withA∩B =∅.
• For a family F ⊂2X and x∈X let
(0.1) F(x)={F \{x}:x∈F ∈F},
(0.2) F(x¯)={F :x(cid:6)∈F ∈F}.
• We say that two families F,G ⊂2X are isomorphic if G is
obtained from F by renaming vertices, that is, if there is a
permutation π on X such that
G ={{π(x):x∈F}:F ∈F}.
In this case we write F ∼=G.
• For a family F ⊂2X, the complement family Fc is defined
by Fc ={X\F :F ∈F}.
• For a family F ⊂ 2X and x ∈ X, the degree of x in F
is defined by deg (x) = #{F ∈ F : x ∈ F ∈ F}. If
F
deg (x)=d for all x∈X, then F is called d-regular.
F
• Avectorxisusuallyconsideredtobeacolumnvector,and
x denotes the ith entry.
i
• For a matrix A, the transposed matrix is denoted by AT.
Theidentitymatrixandtheall-onesmatrixaredenotedby
I and J, respectively. If we need to specify the size, then
I is the n×n identity matrix. Similarly we write 1 and
n
1 for the all-ones vector.
n
• An n-vertex graph G is a graph with n vertices. The adja-
cency matrix A=(a ) of G is an n×n matrix defined by
i,j
a =1if iand j are adjacent inG anda =0otherwise.
i,j i,j
• Let V be a vector space over a field F and let U ⊂V be a
finite subset. Let spanU denote the subspace spanned by
(cid:2)
U, that is, spanU ={ α u:α ∈F}.
u∈U u u
Chapter 1
Introduction
Apart from numbers, sets are the most fundamental notion in math-
ematics. Arguably the simplest sets are finite sets, that is, sets with
a finite number of elements. The present book deals with combina-
torial, mostly extremal problems concerning systems of subsets of a
given finite set. Let us introduce the basic notation first.
For a set X let |X| denote its size, that is, the number of its
elements. If|X|=n,thenX iscalledann-elementset. Forapositive
integer n let [n] = {1,2,...,n} denote the standard n-element set.
The power set 2[n] consists of the 2n subs(cid:3)ets(cid:4)(including [n] and the
empty set ∅) of [n]. For 0 ≤ k ≤ n let [n] denote the collection
k
of all k-element subsets of [n]. A subset F of 2[n] is called a family
of subsets. We can think of F ⊂ 2X as a hypergraph. In this case
the vertex set is X and the edge set is F itself. In this sen(cid:3)se(cid:4)we
sometimes call a member of F an edge of F. A family F ⊂ [n] is
k
called k-uniform (or a k-uniform hypergraph).
To explain the topics in this book, probably it is best to state
and prove Sperner’s Theorem, the first result in this area. We need
some simple notions. If
F (cid:2)F (cid:2)···(cid:2)F
0 1 l
then (F ,...,F ) is called a chain of length l. If for a family F no
0 l
F,G ∈ F satisfy F (cid:2) G, F is called an antichain. The reason is
1