Combinatorics in Hungary and Extremal Set Theory Gyula O.H. Katona - - PowerPoint PPT Presentation

Combinatorics in Hungary and Extremal Set Theory

Gyula O.H. Katona R´ enyi Institute, Budapest Jiao Tong University Colloquium Talk October 22, 2014

Combinatorics in Hungary

A little history.

1

Combinatorics in Hungary

A little history. Turkey occupied half of Hungary in 1526, Austria the other half.

2

Combinatorics in Hungary

A little history. Turkey occupied half of Hungary in 1526, Austria the other half. The united European armies pushed out the Turks around 1670, Hungary became a colony of Austria.

3

Combinatorics in Hungary

A little history. Turkey occupied half of Hungary in 1526, Austria the other half. The united European armies pushed out the Turks around 1670, Hungary became a colony of Austria. Revolution and freedom fight against Austria in 1948, one and a half year long war.

4

Combinatorics in Hungary

A little history. Turkey occupied half of Hungary in 1526, Austria the other half. The united European armies pushed out the Turks around 1670, Hungary became a colony of Austria. Revolution and freedom fight against Austria in 1948, one and a half year long war. Hungary was winning, Austria asked the help of the Russians. The two big countries easily suppressed the revolution.

5

Combinatorics in Hungary

A little history. Turkey occupied half of Hungary in 1526, Austria the other half. The united European armies pushed out the Turks around 1670, Hungary became a colony of Austria. Revolution and freedom fight against Austria in 1848, one and a half year long war. Hungary was winning, Austria asked the help of the Russians. The two big countries easily suppressed the revolution. Agreement in 1867. The Austrian Monarchy became the Austro- Hungarian Monarchy. Very fast economic progress in the Hungarian part.

6

Combinatorics in Hungary

Educational reforms on every level.

7

Combinatorics in Hungary

Educational reforms on every level. General level of elementary schools raised. Talent search. Good math professor at universities. One of them: Gyula K¨

• nig.

8

Combinatorics in Hungary

Educational reforms on every level. General level of elementary schools raised. Talent search. Good math professor at universities. One of them: Gyula K¨

• nig.

The world very first mathematical journal for high school students: 1896!

9

Combinatorics in Hungary

Educational reforms on every level. General level of elementary schools raised. Talent search. Good math professor at universities. One of them: Gyula K¨

• nig.

The world very first mathematical journal for high school students: 1896! The first mathematical contest for high school students in the world in 1920.

10

Combinatorics in Hungary

Educational reforms on every level. General level of elementary schools raised. Talent search. Good math professor at universities. One of them: Gyula K¨

• nig.

The world very first mathematical journal for high school students: 1894! The first mathematical contest for high school students in the world in 1894! Hars´ anyi (NP), Von Neumann, Teller, Wigner (NP) came from the same high school in Budapest. Szil´ ard went to another strong school.

11

Marriage problem = Perfect Matching

Given n girls and n boys, their “knowing each other” is given with a bipartite graph.

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Marriage problem = Perfect Matching

Given n girls and n boys, their “knowing each other” is given with a bipartite graph. Can all the girls find husbands?

13

Marriage problem = Perfect Matching

Given n girls and n boys, their “knowing each other” is given with a bipartite graph. Necessary condition: every set A of girls need to know at least |A| boys.

14

Marriage problem = Perfect Matching

Given n girls and n boys, their “knowing each other” is given with a bipartite graph. Necessary condition: every set A of girls need to know at least |A| boys.

15

Marriage problem = Perfect Matching

Given n girls and n boys, their “knowing each other” is given with a bipartite graph. Necessary condition: every set A of girls need to know at least |A| boys.

16

Marriage problem = Perfect Matching

Given n girls and n boys, their “knowing each other” is given with a bipartite graph. Theorem (D´ enes K¨

• nig, 1916, Hall, 1935) The condition is necessary and

sufficient.

17

First book in Graph Theory by D´ enes K¨

• nig, 1936.

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First book in Graph Theory by D´ enes K¨

• nig, 1936.

Under his influence Erd˝

• s, Gallai, Szekeres, Tur´

an started to think about graphs.

19

First book in Graph Theory by D´ enes K¨

• nig, 1936.

Under his influence Erd˝

• s, Gallai, Szekeres, Tur´

an started to think about graphs. Erd˝

• s and Szekeres found the finite version of Ramsey theorem.

20

First book in Graph Theory by D´ enes K¨

• nig, 1936.

Under his influence Erd˝

• s, Gallai, Szekeres, Tur´

an started to think about graphs. Erd˝

• s and Szekeres found the finite version of Ramsey theorem.

Because of the nazism, Szekeres fled to Shanghai, then to Australia.

21

First book in Graph Theory by D´ enes K¨

• nig, 1936.

Under his influence Erd˝

• s, Gallai, Szekeres, Tur´

an started to think about graphs. Erd˝

• s and Szekeres found the finite version of Ramsey theorem.

Because of the nazism, Szekeres fled to Shanghai, then to Australia. Erd˝

• s went to England and the USA.

22

First book in Graph Theory by D´ enes K¨

• nig, 1936.

Under his influence Erd˝

• s, Gallai, Szekeres, Tur´

an started to think about graphs. Erd˝

• s and Szekeres found the finite version of Ramsey theorem.

Because of the nazism, Szekeres fled to Shanghai, then to Australia. Erd˝

• s went to England and the USA.

Tur´ an was in a labor camp, yet he was doing graph theory there!

23

Erd˝

• s, Ko, Rado

Notation: [n] = {1, 2, . . . , n}. A family F ⊂ 2[n] is intersecting if F ∩ G = ∅ holds for every pair F, G ∈ F.

24

Erd˝

• s, Ko, Rado

Notation: [n] = {1, 2, . . . , n}. A family F ⊂ 2[n] is intersecting if F ∩ G = ∅ holds for every pair F, G ∈ F.

25

Erd˝

• s, Ko, Rado

Theorem (Erd˝

• s – Ko – Rado, done in 1938, published in 1961) If F ⊂

[n]

k

• is intersecting where k ≤ n

2 then

|F| ≤ n − 1 k − 1

• .

26

I found a short proof in 1972. Erd˝

• s said: it is from the BOOK.

27

Erd˝

• s, Ko, Rado

I found a short proof in 1972. Erd˝

• s said: it is from the BOOK.

Martin Aigner and G¨ unter Ziegler: Proofs from THE BOOK. Springer 1998.

28

Erd˝

• s, Ko, Rado

I found a short proof in 1972. Erd˝

• s said: it is from the BOOK.

Martin Aigner and G¨ unter Ziegler: Proofs from THE BOOK. Springer 1998. Erd˝

• s returned to Hungary in 1954.

29

30

31

Alfr´ ed R´ enyi a strong probabilist, director of the Mathematical Institute of the Hungarian Academy of Sciences.

32

33

Random graphs

Alfr´ ed R´ enyi a strong probabilist, director of the Mathematical Institute of the Hungarian Academy of Sciences. Erd˝

• s and R´

enyi started to study RANDOM GRAPHS.

34

Random graphs

Alfr´ ed R´ enyi a strong probabilist, director of the Mathematical Institute of the Hungarian Academy of Sciences. Erd˝

• s and R´

enyi started to study RANDOM GRAPHS. 3 different models: (1) One graph is chosen randomly from the possible n

2

• e
• graphs with e edges.

35

(2) Choose the edges independently with probability p = e n

2

.

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Random graphs

(3) Add edges one by one and look at the graph when e edges are added. They are equivalent for reasonable problems.

37

Random graphs

Theorem The first cycle appears around e = n

2.

Theorem The graph becomes connected around e = 1

2n log n.

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Random graphs

Theorem The first cycle appears around e = n

2.

Theorem The graph becomes connected around e = 1

2n log n.

Before that, “giant component”.

39

Random graphs

Many applications in natural sciences. E.g. physics, social graphs, of course, with other probabilities.

40

Shadows

• Definition. The shadow σ(A) of A is the family of all k − 1-element sets
• btained from the members of A by deleting exactly one element.

σ(A) = {B : |B| = k − 1, B ⊂ A ∈ A}

41

Shadows

• Definition. The shadow σ(A) of A is the family of all k − 1-element sets
• btained from the members of A by deleting exactly one element.

σ(A) = {B : |B| = k − 1, B ⊂ A ∈ A}

42

Shadows

• Definition. The shadow σ(A) of A is the family of all k − 1-element sets
• btained from the members of A by deleting exactly one element.

σ(A) = {B : |B| = k − 1, B ⊂ A ∈ A}

43

Shadows

• Definition. The shadow σ(A) of A is the family of all k − 1-element sets
• btained from the members of A by deleting exactly one element.

σ(A) = {B : |B| = k − 1, B ⊂ A ∈ A}

44

Shadows

• Definition. The shadow σ(A) of A is the family of all k − 1-element sets
• btained from the members of A by deleting exactly one element.

σ(A) = {B : |B| = k − 1, B ⊂ A ∈ A}

45

Shadows

Given n, k and |F|, minimize |σ(F)|. If lucky then |F| = a

k

• holds for an integer a then the best construction is

min|σ(F)| = a

k−1

• 46

• therwise

Lemma If 0 < k, m are integers then one can find integers ak > ak−1 > . . . > at ≥ t ≥ 1 such that m = ak k

• +

ak−1 k − 1

• + . . . +

at t

• and they are unique.

This is called the canonical form of m.

47

• therwise

Shadow Theorem (Kruskal-K, 1960’s) If n, k and |F| are given the canonical form of |F| is |F| = ak k

• +

ak−1 k − 1

• + . . . +

at t

• then

min |σ(F)| = ak k − 1

• +

ak−1 k − 2

• + . . . +

at t − 1

• .

48

Szemer´ edi’s regularity lemma

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Szemer´ edi’s regularity lemma

Large graphs follow some pattern, some rules.

50

Szemer´ edi’s regularity lemma

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Szemer´ edi’s regularity lemma

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Szemer´ edi’s regularity lemma

53

Szemer´ edi’s regularity lemma

Regularity lemma(Szemer´ edi, 1975) A graph with many (many-many) vertices can be partitioned into equally sized subsets in such a way that most pairs of subsets span an ε-regular bipartite graph.

54

A problem motivated by cryptology

A, B ⊂ [n] 3

• , |A| = |B|

σ(A) ∩ σ(B) = ∅ (In other words, if A ∈ A, B ∈ B then |A ∩ B| ≤ 1.) Find f(n, 3) = max |A| under these conditions.

55

Only an estimate

If x is a real number,

x

k

• = x(x−1)...(x−k+1)

k!

.

• Theorem. (Lov´

asz’ version of the Shadow theorem) If A is a family of k- element sets,

|A| = x k

• then

|σ(A)| ≥ x k

• .

56

A weak estimate from Shadow Theorem

Choose x in this way: |A| = |B| = x

3

• By the Shadow Theorem:

x

2

• ≤ |σ(A)|, |σ(B)|

2 x 2

• ≤ |σ(A)| + |σ(B)| ≤

n 2

• From here, asymptotically

|A| = x 3

• ≤

1 2 √ 2 n 3

• (1 + o(1))

57

Trivial construction gives:

|A| = n3 48(1 + o(1)). A =

• (a, b, c) : a < b < c ≤ n

2

• , B =
• (a, b, c) : n

2 ≤ a < b < c

• .

58

A better construction: |A| = n3

24(1 + o(1)).

A =

• (a, b, c) : b+c

2 ≤ n 2

• B = {(a, b, c) : n

2 < a+b 2

• .

59

We have 0.25 ≤ lim sup f(n,3)

(n

3)

≤ 0.35355....

60

We have 0.25 ≤ lim sup f(n,3)

(n

3)

≤ 0.35355....

Theorem (Frankl-Kato-Katona-Tokushige) f(n, 3) = 0.278... n 3

• (1 + o(1))

61

Theorem (Frankl-Kato-Katona-Tokushige) f(n, 3) = κ3 n 3

• (1 + o(1))

where κ is the unique real root in the (0,1)-interval of the equation z3 = (1 − z)3 + 3z(1 − z)2.

62

Theorem (Frankl-Kato-Katona-Tokushige, 2013) A1, A2 ⊂ [n]

k

• , |A1| = |A2|, and

A1 ∈ A1, A2 ∈ A2 imply |A1 ∩ A2| ≤ 1 then max |A| = µk

k

n k

• (1 + o(1))

where µk is the unique real root in the (0,1)-interval of the equation zk = (1 − z)k + kz(1 − z)k−1.

63

Some months later Huang, Linial, Naves, Peled, Sudakov proved a very similar result. They considered two families A ⊂ [n]

k

• , B ⊂

[n]

ℓ

• where A ∈

A, b ∈ B imlies |A ∩ B| ≤ 1. Supposing |A| n

k

= α they asymptotically determine max |B| n

ℓ

. The case k = ℓ gives back our result, but their upper estimate is less sharp.

64

Some months later Huang, Linial, Naves, Peled, Sudakov proved a very similar result. They considered two families A ⊂ [n]

k

• , B ⊂

[n]

ℓ

• where A ∈

A, b ∈ B imlies |A ∩ B| ≤ 1. Supposing |A| n

k

= α they asymptotically determine max |B| n

ℓ

. The case k = ℓ gives back our result, but their upper estimate is less sharp. Their motivation is theoretical.

65

Badly needed generalizations s-shadows (s = k − 2), that is, |A ∩ B| ≤ r.

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Badly needed generalizations s-shadows (s = k − 2), that is, |A ∩ B| ≤ r.

More families rather than only 2.

67

A family F ⊂ 2[n] is called a (u, v)-union-intersecting if for different members F1, . . . , Fu, G1, . . . , Gv the following holds: (∪u

i=1Fi) ∩

• ∪v

j=1Gj

• = ∅.

68

A family F ⊂ 2[n] is called a (u, v)-union-intersecting if for different members F1, . . . , Fu, G1, . . . , Gv the following holds: (∪u

i=1Fi) ∩

• ∪v

j=1Gj

• = ∅.

Theorem (Katona-D.T. Nagy 2014+) Let 1 ≤ u ≤ v and suppose that the family F ⊂ [n]

k

• is a (u, v)-union–intersecting family then

|F| ≤ n − 1 k − 1

• + u − 1

holds if n > n(k, v).

69

A family F ⊂ 2[n] is called a (u, v)-union-intersecting if for different members F1, . . . , Fu, G1, . . . , Gv the following holds: (∪u

i=1Fi) ∩

• ∪v

j=1Gj

• = ∅.

Theorem (Katona-D.T. Nagy 2014+) Let 1 ≤ u ≤ v and suppose that the family F ⊂ [n]

k

• is a (u, v)-union–intersecting family then

|F| ≤ n − 1 k − 1

• + u − 1

holds if n > n(k, v).

70

71

72

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