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The independence numbers and the chromatic numbers of random subgraphs of Knesers graphs and their generalizations Andrei Raigorodskii Moscow Institute of Physics and Technology, Yandex, Moscow, Russia Shanghai, China, 26.03.2017 A.

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SLIDE 1

The independence numbers and the chromatic numbers of random subgraphs of Kneser’s graphs and their generalizations

Andrei Raigorodskii

Moscow Institute of Physics and Technology, Yandex, Moscow, Russia

Shanghai, China, 26.03.2017

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 1 / 11

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SLIDE 2

The Erd˝

  • s–Ko–Rado Theorem

Erd˝

  • s–Ko–Rado, 1961

Let [n] = {1 ; 2 ;

: : : ; n}. Assume that F ⊂

[n]

r
  • with
r n=2 is such a collection
  • f
r-subsets that any two of them intersect. Then |F|
  • n−1
r−1
  • .
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 2 / 11

slide-3
SLIDE 3

The Erd˝

  • s–Ko–Rado Theorem

Erd˝

  • s–Ko–Rado, 1961

Let [n] = {1 ; 2 ;

: : : ; n}. Assume that F ⊂

[n]

r
  • with
r n=2 is such a collection
  • f
r-subsets that any two of them intersect. Then |F|
  • n−1
r−1
  • .

Of course the bound

  • n−1
r−1
  • is attained on a “star”.
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 2 / 11

slide-4
SLIDE 4

The Erd˝

  • s–Ko–Rado Theorem

Erd˝

  • s–Ko–Rado, 1961

Let [n] = {1 ; 2 ;

: : : ; n}. Assume that F ⊂

[n]

r
  • with
r n=2 is such a collection
  • f
r-subsets that any two of them intersect. Then |F|
  • n−1
r−1
  • .

Of course the bound

  • n−1
r−1
  • is attained on a “star”.

Hilton–Milner, 1967 Let [n] = {1 ; 2 ;

: : : ; n}. Assume that F ⊂

[n]

r
  • with
r n=2 is such a collection
  • f
r-subsets that any two of them intersect and F is not a star. Then

|F|

  • n−1
r−1
  • −
  • n−r−1
r−1
  • + 1.
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 2 / 11

slide-5
SLIDE 5

The Erd˝

  • s–Ko–Rado Theorem

Erd˝

  • s–Ko–Rado, 1961

Let [n] = {1 ; 2 ;

: : : ; n}. Assume that F ⊂

[n]

r
  • with
r n=2 is such a collection
  • f
r-subsets that any two of them intersect. Then |F|
  • n−1
r−1
  • .

Of course the bound

  • n−1
r−1
  • is attained on a “star”.

Hilton–Milner, 1967 Let [n] = {1 ; 2 ;

: : : ; n}. Assume that F ⊂

[n]

r
  • with
r n=2 is such a collection
  • f
r-subsets that any two of them intersect and F is not a star. Then

|F|

  • n−1
r−1
  • −
  • n−r−1
r−1
  • + 1.

It’s a famous stability result.

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 2 / 11

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SLIDE 6

The Erd˝

  • s–Ko–Rado Theorem

Erd˝

  • s–Ko–Rado, 1961

Let [n] = {1 ; 2 ;

: : : ; n}. Assume that F ⊂

[n]

r
  • with
r n=2 is such a collection
  • f
r-subsets that any two of them intersect. Then |F|
  • n−1
r−1
  • .

Of course the bound

  • n−1
r−1
  • is attained on a “star”.

Hilton–Milner, 1967 Let [n] = {1 ; 2 ;

: : : ; n}. Assume that F ⊂

[n]

r
  • with
r n=2 is such a collection
  • f
r-subsets that any two of them intersect and F is not a star. Then

|F|

  • n−1
r−1
  • −
  • n−r−1
r−1
  • + 1.

It’s a famous stability result. Other stability results were proposed by Balogh, Bohman, Mubayi et al. using the notion of a random hypergraph.

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 2 / 11

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SLIDE 7

A graph-theoretic point of view

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 3 / 11

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SLIDE 8

A graph-theoretic point of view

The independence number

( G) of a graph G is the maximum number of

pairwise disjoint vertices of

G.
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 3 / 11

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SLIDE 9

A graph-theoretic point of view

The independence number

( G) of a graph G is the maximum number of

pairwise disjoint vertices of

G.

Kneser’s graph

K G n;r = ( V ; E), where V =

[n]

r
  • ,
E = {( A; B) : A ∩ B = ∅} :
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 3 / 11

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SLIDE 10

A graph-theoretic point of view

The independence number

( G) of a graph G is the maximum number of

pairwise disjoint vertices of

G.

Kneser’s graph

K G n;r = ( V ; E), where V =

[n]

r
  • ,
E = {( A; B) : A ∩ B = ∅} :

Erd˝

  • s–Ko–Rado

If

r n=2, then ( K G n;r) =
  • n−1
r−1
  • .
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 3 / 11

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SLIDE 11

A graph-theoretic point of view

The independence number

( G) of a graph G is the maximum number of

pairwise disjoint vertices of

G.

Kneser’s graph

K G n;r = ( V ; E), where V =

[n]

r
  • ,
E = {( A; B) : A ∩ B = ∅} :

Erd˝

  • s–Ko–Rado

If

r n=2, then ( K G n;r) =
  • n−1
r−1
  • .

The chromatic number

( G) of a graph is the smallest number of colors needed

to color all the vertices so that no two vertices of the same color are joined by an edge.

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 3 / 11

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SLIDE 12

A graph-theoretic point of view

The independence number

( G) of a graph G is the maximum number of

pairwise disjoint vertices of

G.

Kneser’s graph

K G n;r = ( V ; E), where V =

[n]

r
  • ,
E = {( A; B) : A ∩ B = ∅} :

Erd˝

  • s–Ko–Rado

If

r n=2, then ( K G n;r) =
  • n−1
r−1
  • .

The chromatic number

( G) of a graph is the smallest number of colors needed

to color all the vertices so that no two vertices of the same color are joined by an edge. Lov´ asz, 1978 If

r n=2, then ( K G n;r) = n − 2 r + 2.
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 3 / 11

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SLIDE 13

Random subgraphs of Kneser’s graphs

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 4 / 11

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SLIDE 14

Random subgraphs of Kneser’s graphs

Let

p ∈ [0 ; 1]. Then K G n;r ;p is obtained from K G n;r by keeping any of the edges
  • f Kneser’s graph with probability
p.
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 4 / 11

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SLIDE 15

Random subgraphs of Kneser’s graphs

Let

p ∈ [0 ; 1]. Then K G n;r ;p is obtained from K G n;r by keeping any of the edges
  • f Kneser’s graph with probability
p.

Bogoliubskiy, Gusev, Pyaderkin, A.M., 2013 If

r 2 is fixed and n → ∞, then w.h.p. ( K G n;r ;1=2) ∼
  • n−1
r−1
  • .
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 4 / 11

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SLIDE 16

Random subgraphs of Kneser’s graphs

Let

p ∈ [0 ; 1]. Then K G n;r ;p is obtained from K G n;r by keeping any of the edges
  • f Kneser’s graph with probability
p.

Bogoliubskiy, Gusev, Pyaderkin, A.M., 2013 If

r 2 is fixed and n → ∞, then w.h.p. ( K G n;r ;1=2) ∼
  • n−1
r−1
  • .

That’s another kind of stability. Moreover

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 4 / 11

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SLIDE 17

Random subgraphs of Kneser’s graphs

Let

p ∈ [0 ; 1]. Then K G n;r ;p is obtained from K G n;r by keeping any of the edges
  • f Kneser’s graph with probability
p.

Bogoliubskiy, Gusev, Pyaderkin, A.M., 2013 If

r 2 is fixed and n → ∞, then w.h.p. ( K G n;r ;1=2) ∼
  • n−1
r−1
  • .

That’s another kind of stability. Moreover Bollob´ as, Narayanan, A.M., 2016 Fix a real number

" > 0 and let r = r( n) be a natural number such that

2

r( n) =
  • (
n1=3). Let p ( n; r) = (( r + 1) log n − r log r) =
  • n−1
r−1
  • . Then as
n → ∞,

P

  • (
K G n;r ;p) =
  • n − 1
r − 1
  • →
  • 1

if

p (1 + ") p ( n; r)

if

p (1 − ") p ( n; r) :
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 4 / 11

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SLIDE 18

Random subgraphs of Kneser’s graphs

Let

p ∈ [0 ; 1]. Then K G n;r ;p is obtained from K G n;r by keeping any of the edges
  • f Kneser’s graph with probability
p.

Bogoliubskiy, Gusev, Pyaderkin, A.M., 2013 If

r 2 is fixed and n → ∞, then w.h.p. ( K G n;r ;1=2) ∼
  • n−1
r−1
  • .

That’s another kind of stability. Moreover Bollob´ as, Narayanan, A.M., 2016 Fix a real number

" > 0 and let r = r( n) be a natural number such that

2

r( n) =
  • (
n1=3). Let p ( n; r) = (( r + 1) log n − r log r) =
  • n−1
r−1
  • . Then as
n → ∞,

P

  • (
K G n;r ;p) =
  • n − 1
r − 1
  • →
  • 1

if

p (1 + ") p ( n; r)

if

p (1 − ") p ( n; r) :

Successively improved by Das and Tran.

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 4 / 11

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SLIDE 19

Random subgraphs of Kneser’s graphs

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 5 / 11

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SLIDE 20

Random subgraphs of Kneser’s graphs

Very simply the chromatic number of

K G n;r is not so stable as the independence

number: w.h.p. even

( K G n;r ;1=2) < n − 2 r + 2 = ( K G n;r) :

However

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 5 / 11

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SLIDE 21

Random subgraphs of Kneser’s graphs

Very simply the chromatic number of

K G n;r is not so stable as the independence

number: w.h.p. even

( K G n;r ;1=2) < n − 2 r + 2 = ( K G n;r) :

However Kupavskii, 2016 For many different

n; r ; p, w.h.p. ( K G n;r ;p) ∼ ( K G n;r) = n − 2 r + 2 :
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 5 / 11

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SLIDE 22

Random subgraphs of Kneser’s graphs

Very simply the chromatic number of

K G n;r is not so stable as the independence

number: w.h.p. even

( K G n;r ;1=2) < n − 2 r + 2 = ( K G n;r) :

However Kupavskii, 2016 For many different

n; r ; p, w.h.p. ( K G n;r ;p) ∼ ( K G n;r) = n − 2 r + 2 :

For example, if

g( n) is any growing function and r is arbitrary in the range

between 2 and

n

2 −

g( n), then for any fixed p, ( K G n;r ;p) ∼ ( K G n;r) :
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 5 / 11

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SLIDE 23

An important generalization

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 6 / 11

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SLIDE 24

An important generalization

Let

G( n; r ; s) be a graph whose vertices are all the r-subsets of [n] and whose

edges are all the vertex pairs with intersection exactly equal to

s.
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 6 / 11

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SLIDE 25

An important generalization

Let

G( n; r ; s) be a graph whose vertices are all the r-subsets of [n] and whose

edges are all the vertex pairs with intersection exactly equal to

s.

For example,

G( n; r ; 0) = K G n;r, and, more specifically, G( n; 1 ; 0) = K n —

complete graph.

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 6 / 11

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SLIDE 26

An important generalization

Let

G( n; r ; s) be a graph whose vertices are all the r-subsets of [n] and whose

edges are all the vertex pairs with intersection exactly equal to

s.

For example,

G( n; r ; 0) = K G n;r, and, more specifically, G( n; 1 ; 0) = K n —

complete graph. Other

G( n; r ; s) are well motivated by coding theory, Ramsey theory as well

Borsuk’s and Nelson–Hadwiger’s problems in combinatorial geometry.

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 6 / 11

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SLIDE 27

An important generalization

Let

G( n; r ; s) be a graph whose vertices are all the r-subsets of [n] and whose

edges are all the vertex pairs with intersection exactly equal to

s.

For example,

G( n; r ; 0) = K G n;r, and, more specifically, G( n; 1 ; 0) = K n —

complete graph. Other

G( n; r ; s) are well motivated by coding theory, Ramsey theory as well

Borsuk’s and Nelson–Hadwiger’s problems in combinatorial geometry. Many results concerning

( G( n; r ; s)). The most important ones follow.
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 6 / 11

slide-28
SLIDE 28

An important generalization

Let

G( n; r ; s) be a graph whose vertices are all the r-subsets of [n] and whose

edges are all the vertex pairs with intersection exactly equal to

s.

For example,

G( n; r ; 0) = K G n;r, and, more specifically, G( n; 1 ; 0) = K n —

complete graph. Other

G( n; r ; s) are well motivated by coding theory, Ramsey theory as well

Borsuk’s and Nelson–Hadwiger’s problems in combinatorial geometry. Many results concerning

( G( n; r ; s)). The most important ones follow.

Frankl, F¨ uredi, 1985 For any fixed

r ; s, there exist ( r ; s) ; d( r ; s) such that ( r ; s) nmax{s;r−s−1} ( G( n; r ; s)) d( r ; s) nmax{s;r−s−1} :
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 6 / 11

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SLIDE 29

An important generalization

Let

G( n; r ; s) be a graph whose vertices are all the r-subsets of [n] and whose

edges are all the vertex pairs with intersection exactly equal to

s.

For example,

G( n; r ; 0) = K G n;r, and, more specifically, G( n; 1 ; 0) = K n —

complete graph. Other

G( n; r ; s) are well motivated by coding theory, Ramsey theory as well

Borsuk’s and Nelson–Hadwiger’s problems in combinatorial geometry. Many results concerning

( G( n; r ; s)). The most important ones follow.

Frankl, F¨ uredi, 1985 For any fixed

r ; s, there exist ( r ; s) ; d( r ; s) such that ( r ; s) nmax{s;r−s−1} ( G( n; r ; s)) d( r ; s) nmax{s;r−s−1} :

Frankl, F¨ uredi, 1985 For any fixed

r ; s such that r > 2 s + 1, ( G( n; r ; s)) =
  • n −
s − 1 r − s − 1
  • = Θ
  • n
r−s−1 :
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 6 / 11

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SLIDE 30

Main result

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 7 / 11

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SLIDE 31

Main result

We consider

G p( n; r ; s) — random subgraphs of the graphs G( n; r ; s).
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 7 / 11

slide-32
SLIDE 32

Main result

We consider

G p( n; r ; s) — random subgraphs of the graphs G( n; r ; s).

Pyaderkin, A.M., 2016 Let

r ; s be fixed and " > 0. There exists a Æ = Æ( r ; s; ") such that w.h.p. ( G1=2( n; r ; s)) (1 + ") ( G( n; r ; s)) + Æ
  • n
s
  • log2
n:
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 7 / 11

slide-33
SLIDE 33

Main result

We consider

G p( n; r ; s) — random subgraphs of the graphs G( n; r ; s).

Pyaderkin, A.M., 2016 Let

r ; s be fixed and " > 0. There exists a Æ = Æ( r ; s; ") such that w.h.p. ( G1=2( n; r ; s)) (1 + ") ( G( n; r ; s)) + Æ
  • n
s
  • log2
n:

Let

r 2 s + 1. Then due to Frankl and F¨

uredi we have

( G( n; r ; s)) = Θ( n s).
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 7 / 11

slide-34
SLIDE 34

Main result

We consider

G p( n; r ; s) — random subgraphs of the graphs G( n; r ; s).

Pyaderkin, A.M., 2016 Let

r ; s be fixed and " > 0. There exists a Æ = Æ( r ; s; ") such that w.h.p. ( G1=2( n; r ; s)) (1 + ") ( G( n; r ; s)) + Æ
  • n
s
  • log2
n:

Let

r 2 s + 1. Then due to Frankl and F¨

uredi we have

( G( n; r ; s)) = Θ( n s).

At the same time,

  • n
s
  • = Θ(
n s). Thus, w.h.p. we have ( G1=2( n; r ; s)) = O( n s log2 n).
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 7 / 11

slide-35
SLIDE 35

Main result

We consider

G p( n; r ; s) — random subgraphs of the graphs G( n; r ; s).

Pyaderkin, A.M., 2016 Let

r ; s be fixed and " > 0. There exists a Æ = Æ( r ; s; ") such that w.h.p. ( G1=2( n; r ; s)) (1 + ") ( G( n; r ; s)) + Æ
  • n
s
  • log2
n:

Let

r 2 s + 1. Then due to Frankl and F¨

uredi we have

( G( n; r ; s)) = Θ( n s).

At the same time,

  • n
s
  • = Θ(
n s). Thus, w.h.p. we have ( G1=2( n; r ; s)) = O( n s log2 n).

One can easily show using the first moment method that w.h.p.

( G1=2( n; r ; s)) = Ω( n s log2 n), which means that w.h.p. ( G1=2( n; r ; s)) = Θ( n s log2 n) ; r 2 s + 1 :
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 7 / 11

slide-36
SLIDE 36

Main result

We consider

G p( n; r ; s) — random subgraphs of the graphs G( n; r ; s).

Pyaderkin, A.M., 2016 Let

r ; s be fixed and " > 0. There exists a Æ = Æ( r ; s; ") such that w.h.p. ( G1=2( n; r ; s)) (1 + ") ( G( n; r ; s)) + Æ
  • n
s
  • log2
n:

Let

r 2 s + 1. Then due to Frankl and F¨

uredi we have

( G( n; r ; s)) = Θ( n s).

At the same time,

  • n
s
  • = Θ(
n s). Thus, w.h.p. we have ( G1=2( n; r ; s)) = O( n s log2 n).

One can easily show using the first moment method that w.h.p.

( G1=2( n; r ; s)) = Ω( n s log2 n), which means that w.h.p. ( G1=2( n; r ; s)) = Θ( n s log2 n) ; r 2 s + 1 :

By the way, this agrees perfectly with the results concerning

G( n; p) = G p( n; 1 ; 0).
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 7 / 11

slide-37
SLIDE 37

Main result: continuation

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 8 / 11

slide-38
SLIDE 38

Main result: continuation

Now let

r > 2 s + 1. Once again, Frankl and F¨

uredi tell us that

( G( n; r ; s)) =
  • n −
s − 1 r − s − 1
  • :
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 8 / 11

slide-39
SLIDE 39

Main result: continuation

Now let

r > 2 s + 1. Once again, Frankl and F¨

uredi tell us that

( G( n; r ; s)) =
  • n −
s − 1 r − s − 1
  • :

But this time

  • n
s
  • =
  • n−s−1
r−s−1
  • , so that we get w.h.p.
( G1=2( n; r ; s)) (1 +
  • (1))
  • n −
s − 1 r − s − 1
  • ;

which means that w.h.p.

( G1=2( n; r ; s)) ∼ ( G( n; r ; s)) :

Asymptotic stability!

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 8 / 11

slide-40
SLIDE 40

Main result: continuation

Now let

r > 2 s + 1. Once again, Frankl and F¨

uredi tell us that

( G( n; r ; s)) =
  • n −
s − 1 r − s − 1
  • :

But this time

  • n
s
  • =
  • n−s−1
r−s−1
  • , so that we get w.h.p.
( G1=2( n; r ; s)) (1 +
  • (1))
  • n −
s − 1 r − s − 1
  • ;

which means that w.h.p.

( G1=2( n; r ; s)) ∼ ( G( n; r ; s)) :

Asymptotic stability! Local conclusion If

r 2 s + 1, then the independence number of the random graph G1=2( n; r ; s)

behaves like the independence number of the Erd˝

  • s–R´

enyi random graph: w.h.p. it increases log times when compared to the initial idependence number. Otherwise, it is stable like its analog for Kneser’s graph.

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 8 / 11

slide-41
SLIDE 41

Main result: more stability?

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 9 / 11

slide-42
SLIDE 42

Main result: more stability?

For Kneser’s graphs, we had complete stability. However, for other

r ; s with r > 2 s + 1, we got only asymptotic stability. Is it essential or just a technical

problem?

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 9 / 11

slide-43
SLIDE 43

Main result: more stability?

For Kneser’s graphs, we had complete stability. However, for other

r ; s with r > 2 s + 1, we got only asymptotic stability. Is it essential or just a technical

problem? For

s = 0, we had Hilton–Milner theorem that roughly told us: “If an independent

set is not a star, then it is many times smaller than the stars of maximal sizes.”

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 9 / 11

slide-44
SLIDE 44

Main result: more stability?

For Kneser’s graphs, we had complete stability. However, for other

r ; s with r > 2 s + 1, we got only asymptotic stability. Is it essential or just a technical

problem? For

s = 0, we had Hilton–Milner theorem that roughly told us: “If an independent

set is not a star, then it is many times smaller than the stars of maximal sizes.” Now, we don’t have such results. Moreover, they are not true! Let’s take

G( n; 4 ; 1). The Frankl and Wilson linear algebra method gives the bound ( G( n; 4 ; 1))
  • n

2

  • ∼
n2

2

:
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 9 / 11

slide-45
SLIDE 45

Main result: more stability?

For Kneser’s graphs, we had complete stability. However, for other

r ; s with r > 2 s + 1, we got only asymptotic stability. Is it essential or just a technical

problem? For

s = 0, we had Hilton–Milner theorem that roughly told us: “If an independent

set is not a star, then it is many times smaller than the stars of maximal sizes.” Now, we don’t have such results. Moreover, they are not true! Let’s take

G( n; 4 ; 1). The Frankl and Wilson linear algebra method gives the bound ( G( n; 4 ; 1))
  • n

2

  • ∼
n2

2

:

On the other hand, there are two completely different constructions of independent sets with cardinality of order

n2.
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 9 / 11

slide-46
SLIDE 46

Main result: more stability!

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 10 / 11

slide-47
SLIDE 47

Main result: more stability!

Construction 1 is just a kind of a star: fix 2 elements of [

n] and take all the

4-tuples that contain them. Here we have ∼

n2

2 sets.

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 10 / 11

slide-48
SLIDE 48

Main result: more stability!

Construction 1 is just a kind of a star: fix 2 elements of [

n] and take all the

4-tuples that contain them. Here we have ∼

n2

2 sets.

Construction 2 is as follows. Divide [n] into consecutive

  • n

2

  • pairs of elements.

Then take all the 4-tuples formed by any two such pairs. This way we get ∼

n2

8

sets.

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 10 / 11

slide-49
SLIDE 49

Main result: more stability!

Construction 1 is just a kind of a star: fix 2 elements of [

n] and take all the

4-tuples that contain them. Here we have ∼

n2

2 sets.

Construction 2 is as follows. Divide [n] into consecutive

  • n

2

  • pairs of elements.

Then take all the 4-tuples formed by any two such pairs. This way we get ∼

n2

8

sets. And one can combine the two constructions!

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 10 / 11

slide-50
SLIDE 50

Main result: more stability!

Construction 1 is just a kind of a star: fix 2 elements of [

n] and take all the

4-tuples that contain them. Here we have ∼

n2

2 sets.

Construction 2 is as follows. Divide [n] into consecutive

  • n

2

  • pairs of elements.

Then take all the 4-tuples formed by any two such pairs. This way we get ∼

n2

8

sets. And one can combine the two constructions! Nevertheless

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 10 / 11

slide-51
SLIDE 51

Main result: more stability!

Construction 1 is just a kind of a star: fix 2 elements of [

n] and take all the

4-tuples that contain them. Here we have ∼

n2

2 sets.

Construction 2 is as follows. Divide [n] into consecutive

  • n

2

  • pairs of elements.

Then take all the 4-tuples formed by any two such pairs. This way we get ∼

n2

8

sets. And one can combine the two constructions! Nevertheless Pyaderkin, A.M., 2016 Let

r > 3 be fixed. Then w.h.p. ( G1=2( n; r ; 1)) = ( G( n; r ; 1)) :
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 10 / 11

slide-52
SLIDE 52

Main result: more stability!

Construction 1 is just a kind of a star: fix 2 elements of [

n] and take all the

4-tuples that contain them. Here we have ∼

n2

2 sets.

Construction 2 is as follows. Divide [n] into consecutive

  • n

2

  • pairs of elements.

Then take all the 4-tuples formed by any two such pairs. This way we get ∼

n2

8

sets. And one can combine the two constructions! Nevertheless Pyaderkin, A.M., 2016 Let

r > 3 be fixed. Then w.h.p. ( G1=2( n; r ; 1)) = ( G( n; r ; 1)) :

It is very important to emphasize here that the exact value of

( G( n; r ; 1)) is

unknown for all values of

r!
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 10 / 11

slide-53
SLIDE 53

One more graph

  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 11 / 11

slide-54
SLIDE 54

One more graph

Theorem (Nagy, 1972). If

n ≡ 0 (mod 4), then ( G( n; 3 ; 1)) =
  • n. If
n ≡ 1 (mod 4), then ( G( n; 3 ; 1)) = n − 1. If n ≡ 2 ; 3 (mod 4), then ( G( n; 3 ; 1)) = n − 2.
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 11 / 11

slide-55
SLIDE 55

One more graph

Theorem (Nagy, 1972). If

n ≡ 0 (mod 4), then ( G( n; 3 ; 1)) =
  • n. If
n ≡ 1 (mod 4), then ( G( n; 3 ; 1)) = n − 1. If n ≡ 2 ; 3 (mod 4), then ( G( n; 3 ; 1)) = n − 2.

Theorem (Balogh, Kostochka, A.M., 2012). If

n = 2 k, then ( G( n; 3 ; 1)) = ( n − 1)( n − 2) =6.
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 11 / 11

slide-56
SLIDE 56

One more graph

Theorem (Nagy, 1972). If

n ≡ 0 (mod 4), then ( G( n; 3 ; 1)) =
  • n. If
n ≡ 1 (mod 4), then ( G( n; 3 ; 1)) = n − 1. If n ≡ 2 ; 3 (mod 4), then ( G( n; 3 ; 1)) = n − 2.

Theorem (Balogh, Kostochka, A.M., 2012). If

n = 2 k, then ( G( n; 3 ; 1)) = ( n − 1)( n − 2) =6.

Theorem (Pyaderkin, A.M., 2016). W.h.p.

( G1=2( n; 3 ; 1)) ∼ 2 n log2 n:
  • A. Raigorodskii (MIPT, YND)

2017 Shanghai 11 / 11