Combinatorics in Hungary and Extremal Set...

Combinatorics in Hungaryand Extremal Set Theory

Gyula O.H. KatonaRenyi Institute, Budapest

Jiao Tong University

Colloquium Talk

October 22, 2014

Combinatorics in Hungary

A little history.

1


A little history.

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

2


A little history.


The united European armies pushed out the Turks around 1670, Hungarybecame a colony of Austria.

3


A little history.



Revolution and freedom fight against Austria in 1948, one and a half yearlong war.

4


A little history.



Revolution and freedom fight against Austria in 1948, one and a half yearlong war. Hungary was winning, Austria asked the help of the Russians. Thetwo big countries easily suppressed the revolution.

5


A little history.



Revolution and freedom fight against Austria in 1848, one and a half yearlong war. Hungary was winning, Austria asked the help of the Russians. Thetwo 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


Educational reforms on every level.

7



General level of elementary schools raised. Talent search. Good mathprofessor at universities. One of them: Gyula Konig.

8




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

9





The first mathematical contest for high school students in the world in 1920.

10





The first mathematical contest for high school students in the world in 1894!

Harsanyi (NP), Von Neumann, Teller, Wigner (NP) came from the same highschool in Budapest. Szilard went to another strong school.

11

Marriage problem = Perfect Matching

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

12



Can all the girls find husbands?

13



Necessary condition: every set A of girls need to know at least |A| boys.

14




15




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Theorem (Denes Konig, 1916, Hall, 1935) The condition is necessary andsufficient.

17

First book in Graph Theory by Denes Konig, 1936.

18


Under his influence Erdos, Gallai, Szekeres, Turan started to think aboutgraphs.

19



Erdos and Szekeres found the finite version of Ramsey theorem.

20




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

21





Erdos went to England and the USA.

22





Erdos went to England and the USA.

Turan was in a labor camp, yet he was doing graph theory there!

23

Erdos, Ko, Rado

Notation: [n] = {1, 2, . . . , n}.

A family F ⊂ 2[n] is intersecting if F ∩G 6= ∅ holds for every pair F,G ∈ F .

24

Erdos, Ko, Rado

Notation: [n] = {1, 2, . . . , n}.

A family F ⊂ 2[n] is intersecting if F ∩G 6= ∅ holds for every pair F,G ∈ F .

25

Erdos, Ko, Rado

Theorem (Erdos – 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. Erdos said: it is from the BOOK.

27

Erdos, Ko, Rado


Martin Aigner and Gunter Ziegler: Proofs from THE BOOK. Springer 1998.

28

Erdos, Ko, Rado


Martin Aigner and Gunter Ziegler: Proofs from THE BOOK. Springer 1998.

Erdos returned to Hungary in 1954.

29

Alfred Renyi a strong probabilist, director of the Mathematical Institute ofthe Hungarian Academy of Sciences.

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Random graphs


Erdos and Renyi started to study RANDOM GRAPHS.

34

Random graphs


Erdos and Renyi started to study RANDOM GRAPHS.

3 different models:

(1) One graph is chosen randomly from the possible

((n2

)e

)

graphs with e edges.

35

(2) Choose the edges independently with probability

p =e(n2

).

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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 = n2 .

Theorem The graph becomes connected around e = 12n log n.

38

Random graphs

Theorem The first cycle appears around e = n2 .

Theorem The graph becomes connected around e = 12n 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) ofA is the family of all k − 1-element setsobtained from the members ofA by deleting exactly one element.

σ(A) = {B : |B| = k − 1, B ⊂ A ∈ A}

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Shadows


σ(A) = {B : |B| = k − 1, B ⊂ A ∈ A}

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Shadows


σ(A) = {B : |B| = k − 1, B ⊂ A ∈ A}

43

Shadows


σ(A) = {B : |B| = k − 1, B ⊂ A ∈ A}

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Shadows


σ(A) = {B : |B| = k − 1, B ⊂ A ∈ A}

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Shadows

Given n, k and |F|, minimize |σ(F)|.

If lucky then |F| =(ak

)holds for an integer a then the best construction is

min|σ(F)| =(

ak−1

)

46

otherwise

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

m =

(akk

)+

(ak−1

k − 1

)+ . . .+

(att

)and they are unique.

This is called the canonical form of m.

47

otherwise

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

|F| =(akk

)+

(ak−1

k − 1

)+ . . .+

(att

)

thenmin |σ(F)| =

(akk − 1

)+

(ak−1

k − 2

)+ . . .+

(att− 1

).

48

Szemeredi’s regularity lemma

49


Large graphs follow some pattern, some rules.

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Regularity lemma(Szemeredi, 1975)

A graph with many (many-many) vertices can be partitioned into equallysized subsets in such a way that most pairs of subsets span an ε-regularbipartite 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,(xk

)= x(x−1)...(x−k+1)

k! .

Theorem. (Lovasz’ 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| =(x3

)By the Shadow Theorem:

(x2

)≤ |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 ≤n2

}

{B = {(a, b, c) : n

2 <a+b2

}.

59

We have 0.25 ≤ lim sup f(n,3)

(n3)≤ 0.35355....

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We have 0.25 ≤ lim sup f(n,3)

(n3)≤ 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| = µkk

(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 verysimilar result. They considered two families A ⊂

([n]k

),B ⊂

([n]`

)where A ∈

A, b ∈ B imlies |A ∩B| ≤ 1. Supposing

|A|(nk

) = α

they asymptotically determine

max|B|(n`

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

64

Some months later Huang, Linial, Naves, Peled, Sudakov proved a verysimilar result. They considered two families A ⊂

([n]k

),B ⊂

([n]`

)where A ∈

A, b ∈ B imlies |A ∩B| ≤ 1. Supposing

|A|(nk

) = α

they asymptotically determine

max|B|(n`

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

Their motivation is theoretical.

65

Badly needed generalizations

s-shadows (s 6= k − 2), that is, |A ∩B| ≤ r.

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Badly needed generalizations

s-shadows (s 6= k − 2), that is, |A ∩B| ≤ r.

More families rather than only 2.

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

(∪ui=1Fi) ∩(∪vj=1Gj

)6= ∅.

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)6= ∅.

Theorem (Katona-D.T. Nagy 2014+) Let 1 ≤ u ≤ v and suppose that thefamily F ⊂

([n]k

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

|F| ≤(n− 1

k − 1

)+ u− 1

holds if n > n(k, v).

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)6= ∅.

Theorem (Katona-D.T. Nagy 2014+) Let 1 ≤ u ≤ v and suppose that thefamily F ⊂

([n]k

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

|F| ≤(n− 1

k − 1

)+ u− 1

holds if n > n(k, v).

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