[PPT] - The indep endene numb ers and the hromati numb ers of PowerPoint Presentation

SLIDE 1 The indep enden e numb ers and the hromati numb ers

f

random subgraphs Andrei Raigo ro dskii Mos o w Institute

f

Physi s and T e hnology Mos o w, Russia A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 1 / 10

SLIDE 2 Main question A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 2 / 10

SLIDE 3 Main question Erd

sR
enyi

random graph Let n ∈ N, p ∈ [0, 1]. G(n, p) is

btained

b y dra wing indep endently edges

n n

verti es, ea h with p robabilit y p . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 2 / 10

SLIDE 4 Main question Erd

sR
enyi

random graph Let n ∈ N, p ∈ [0, 1]. G(n, p) is

btained

b y dra wing indep endently edges

n n

verti es, ea h with p robabilit y p . Theo rem Let p b e a

nstant
r

a fun tion tending to zero and b

unded

from b elo w b y a value c

n

, where c > 1 . Let d =

1 1−p

. Then w.h.p.

α(G(n, p)) ∼ 2 logd(np), χ(G(n, p)) ∼ n 2 logd(np).

A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 2 / 10

SLIDE 5 Main question Erd

sR
enyi

random graph Let n ∈ N, p ∈ [0, 1]. G(n, p) is

btained

b y dra wing indep endently edges

n n

verti es, ea h with p robabilit y p . Theo rem Let p b e a

nstant
r

a fun tion tending to zero and b

unded

from b elo w b y a value c

n

, where c > 1 . Let d =

1 1−p

. Then w.h.p.

α(G(n, p)) ∼ 2 logd(np), χ(G(n, p)) ∼ n 2 logd(np).

A general random subgraph Let n ∈ N, p ∈ [0, 1], Gn = (Vn, En)

an

a rbitra ry sequen e

f
graphs. Gn,p

is

btained

from Gn b y k eeping indep endently edges

f Gn

, ea h with p robabilit y p . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 2 / 10

SLIDE 6 Main question Erd

sR
enyi

random graph Let n ∈ N, p ∈ [0, 1]. G(n, p) is

btained

b y dra wing indep endently edges

n n

verti es, ea h with p robabilit y p . Theo rem Let p b e a

nstant
r

a fun tion tending to zero and b

unded

from b elo w b y a value c

n

, where c > 1 . Let d =

1 1−p

. Then w.h.p.

α(G(n, p)) ∼ 2 logd(np), χ(G(n, p)) ∼ n 2 logd(np).

A general random subgraph Let n ∈ N, p ∈ [0, 1], Gn = (Vn, En)

an

a rbitra ry sequen e

f
graphs. Gn,p

is

btained

from Gn b y k eeping indep endently edges

f Gn

, ea h with p robabilit y p . What an b e said ab

ut α(Gn,p)

and χ(Gn,p) ? A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 2 / 10

SLIDE 7 A sp e ial ase Main denition Let r, s, n ∈ N, s < r < n , and let G(n, r, s) = (V, E) , where

V = {x = (x1, . . . , xn) : xi ∈ {0, 1}, x1 + . . . + xn = r}, E = {{x, y} : (x, y) = s}.

A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 3 / 10

SLIDE 8 A sp e ial ase Main denition Let r, s, n ∈ N, s < r < n , and let G(n, r, s) = (V, E) , where

V = {x = (x1, . . . , xn) : xi ∈ {0, 1}, x1 + . . . + xn = r}, E = {{x, y} : (x, y) = s}.

Equivalent denition Let r, s, n ∈ N, s < r < n . Let [n] b e an n -element set, and let

G(n, r, s) = (V, E)

, where

V = [n] r

,

E = {A, B ∈ V : |A ∩ B| = s}.

A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 3 / 10

SLIDE 9 A sp e ial ase Main denition Let r, s, n ∈ N, s < r < n , and let G(n, r, s) = (V, E) , where

V = {x = (x1, . . . , xn) : xi ∈ {0, 1}, x1 + . . . + xn = r}, E = {{x, y} : (x, y) = s}.

Equivalent denition Let r, s, n ∈ N, s < r < n . Let [n] b e an n -element set, and let

G(n, r, s) = (V, E)

, where

V = [n] r

,

E = {A, B ∈ V : |A ∩ B| = s}.

Again, what an b e said ab

ut α(Gp(n, r, s))

and χ(Gp(n, r, s)) ? A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 3 / 10

SLIDE 10 Some motivation Why studying G(n, r, s) ? A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 4 / 10

SLIDE 11 Some motivation Why studying G(n, r, s) ? Co ding theo ry (Johnson's graphs): A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 4 / 10

SLIDE 12 Some motivation Why studying G(n, r, s) ? Co ding theo ry (Johnson's graphs): the indep enden e numb er α(G(n, r, s)) stands fo r the maximum size

f

a

de

with

ne

fo rbidden distan e; A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 4 / 10

SLIDE 13 Some motivation Why studying G(n, r, s) ? Co ding theo ry (Johnson's graphs): the indep enden e numb er α(G(n, r, s)) stands fo r the maximum size

f

a

de

with

ne

fo rbidden distan e; the lique numb er ω(G(4k, 2k, k)) is resp

nsible

fo r the existen e

f

an Hadama rd matrix; et . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 4 / 10

SLIDE 14 Some motivation Why studying G(n, r, s) ? Co ding theo ry (Johnson's graphs): the indep enden e numb er α(G(n, r, s)) stands fo r the maximum size

f

a

de

with

ne

fo rbidden distan e; the lique numb er ω(G(4k, 2k, k)) is resp

nsible

fo r the existen e

f

an Hadama rd matrix; et . Combinato rial geometry: G(n, r, s) is a

distan e

graph, i.e., its edges a re

f

the same length

2(r − s)

. The hromati numb er χ(G(n, r, s)) p rovides imp

rtant

b

unds

in the NelsonHadwiger p roblems

f

spa e

lo

ring as w ell as in the Bo rsuk p roblem

f

pa rtitioning sets in spa es into pa rts

f

smaller diameter. A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 4 / 10

SLIDE 15 Some motivation Why studying G(n, r, s) ? Co ding theo ry (Johnson's graphs): the indep enden e numb er α(G(n, r, s)) stands fo r the maximum size

f

a

de

with

ne

fo rbidden distan e; the lique numb er ω(G(4k, 2k, k)) is resp

nsible

fo r the existen e

f

an Hadama rd matrix; et . Combinato rial geometry: G(n, r, s) is a

distan e

graph, i.e., its edges a re

f

the same length

2(r − s)

. The hromati numb er χ(G(n, r, s)) p rovides imp

rtant

b

unds

in the NelsonHadwiger p roblems

f

spa e

lo

ring as w ell as in the Bo rsuk p roblem

f

pa rtitioning sets in spa es into pa rts

f

smaller diameter.

G(n, r, 0)

is the lassi al Kneser graph; G(n, 1, 0) is just a

mplete

graph. A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 4 / 10

SLIDE 16 Some motivation Why studying G(n, r, s) ? Co ding theo ry (Johnson's graphs): the indep enden e numb er α(G(n, r, s)) stands fo r the maximum size

f

a

de

with

ne

fo rbidden distan e; the lique numb er ω(G(4k, 2k, k)) is resp

nsible

fo r the existen e

f

an Hadama rd matrix; et . Combinato rial geometry: G(n, r, s) is a

distan e

graph, i.e., its edges a re

f

the same length

2(r − s)

. The hromati numb er χ(G(n, r, s)) p rovides imp

rtant

b

unds

in the NelsonHadwiger p roblems

f

spa e

lo

ring as w ell as in the Bo rsuk p roblem

f

pa rtitioning sets in spa es into pa rts

f

smaller diameter.

G(n, r, 0)

is the lassi al Kneser graph; G(n, 1, 0) is just a

mplete

graph. Constru tive b

unds

fo r Ramsey numb ers. A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 4 / 10

SLIDE 17 Random subgraphs

f G(n, r, s) :

indep enden e numb ers Theo rem (F rankl, F uredi, 1985) Let r, s b e xed as n → ∞. If r 2s + 1 , then α(G(n, r, s)) = Θ (ns) . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 5 / 10

SLIDE 18 Random subgraphs

f G(n, r, s) :

indep enden e numb ers Theo rem (F rankl, F uredi, 1985) Let r, s b e xed as n → ∞. If r 2s + 1 , then α(G(n, r, s)) = Θ (ns) . If r > 2s + 1 , then α(G(n, r, s)) = Θ

nr−s−1

. A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 5 / 10

SLIDE 19 Random subgraphs

f G(n, r, s) :

indep enden e numb ers Theo rem (F rankl, F uredi, 1985) Let r, s b e xed as n → ∞. If r 2s + 1 , then α(G(n, r, s)) = Θ (ns) . If r > 2s + 1 , then α(G(n, r, s)) = Θ

nr−s−1

. Theo rem (Bogoliubskiy , Gusev, Py aderkin, A.M., 20132016) Let r, s b e xed as n → ∞. If r 2s + 1 , then w.h.p. α(G1/2(n, r, s)) = Θ (α(G(n, r, s)) log n) . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 5 / 10

SLIDE 20 Random subgraphs

f G(n, r, s) :

indep enden e numb ers Theo rem (F rankl, F uredi, 1985) Let r, s b e xed as n → ∞. If r 2s + 1 , then α(G(n, r, s)) = Θ (ns) . If r > 2s + 1 , then α(G(n, r, s)) = Θ

nr−s−1

. Theo rem (Bogoliubskiy , Gusev, Py aderkin, A.M., 20132016) Let r, s b e xed as n → ∞. If r 2s + 1 , then w.h.p. α(G1/2(n, r, s)) = Θ (α(G(n, r, s)) log n) . If r > 2s + 1 , then w.h.p. α(G1/2(n, r, s)) ∼ α(G(n, r, s)) . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 5 / 10

SLIDE 21 Random subgraphs

f G(n, r, s) :

indep enden e numb ers fo r r > 2s + 1 Let r 2 , s = 0 . Then G(n, r, s) is Kneser's graph. A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 6 / 10

SLIDE 22 Random subgraphs

f G(n, r, s) :

indep enden e numb ers fo r r > 2s + 1 Let r 2 , s = 0 . Then G(n, r, s) is Kneser's graph. Bollob

as,

Na ra y anan, A.M., 2016 Fix a real numb er ε > 0 and let r = r(n) b e a natural numb er su h that

2 r(n) = o(n1/3)

. Let pc(n, r) = ((r + 1) log n − r log r)/

n−1

r−1

.

As n → ∞,

P

α(Gp(n, r, 0)) = α(G(n, r, 0)) =

n − 1 r − 1

→
1

if p (1 + ε)pc(n, r) if p (1 − ε)pc(n, r). A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 6 / 10

SLIDE 23 Random subgraphs

f G(n, r, s) :

indep enden e numb ers fo r r > 2s + 1 Let r 2 , s = 0 . Then G(n, r, s) is Kneser's graph. Bollob

as,

Na ra y anan, A.M., 2016 Fix a real numb er ε > 0 and let r = r(n) b e a natural numb er su h that

2 r(n) = o(n1/3)

. Let pc(n, r) = ((r + 1) log n − r log r)/

n−1

r−1

.

As n → ∞,

P

α(Gp(n, r, 0)) = α(G(n, r, 0)) =

n − 1 r − 1

→
1

if p (1 + ε)pc(n, r) if p (1 − ε)pc(n, r). Su essively imp roved b y Das, T ran, Balogh, and

thers.

A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 6 / 10

SLIDE 24 Random subgraphs

f G(n, r, s) :

indep enden e numb ers fo r r > 2s + 1 Let r 2 , s = 0 . Then G(n, r, s) is Kneser's graph. Bollob

as,

Na ra y anan, A.M., 2016 Fix a real numb er ε > 0 and let r = r(n) b e a natural numb er su h that

2 r(n) = o(n1/3)

. Let pc(n, r) = ((r + 1) log n − r log r)/

n−1

r−1

.

As n → ∞,

P

α(Gp(n, r, 0)) = α(G(n, r, 0)) =

n − 1 r − 1

→
1

if p (1 + ε)pc(n, r) if p (1 − ε)pc(n, r). Su essively imp roved b y Das, T ran, Balogh, and

thers.

Let r 4 , s = 1 . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 6 / 10

SLIDE 25 Random subgraphs

f G(n, r, s) :

indep enden e numb ers fo r r > 2s + 1 Let r 2 , s = 0 . Then G(n, r, s) is Kneser's graph. Bollob

as,

Na ra y anan, A.M., 2016 Fix a real numb er ε > 0 and let r = r(n) b e a natural numb er su h that

2 r(n) = o(n1/3)

. Let pc(n, r) = ((r + 1) log n − r log r)/

n−1

r−1

.

As n → ∞,

P

α(Gp(n, r, 0)) = α(G(n, r, 0)) =

n − 1 r − 1

→
1

if p (1 + ε)pc(n, r) if p (1 − ε)pc(n, r). Su essively imp roved b y Das, T ran, Balogh, and

thers.

Let r 4 , s = 1 . Py aderkin, A.M., 2017 W.h.p. α(G1/2(n, r, s)) = α(G(n, r, s)) . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 6 / 10

SLIDE 26 Random subgraphs

f G(n, r, s) :

indep enden e numb ers fo r r > 2s + 1 Let r 2 , s = 0 . Then G(n, r, s) is Kneser's graph. Bollob

as,

Na ra y anan, A.M., 2016 Fix a real numb er ε > 0 and let r = r(n) b e a natural numb er su h that

2 r(n) = o(n1/3)

. Let pc(n, r) = ((r + 1) log n − r log r)/

n−1

r−1

.

As n → ∞,

P

α(Gp(n, r, 0)) = α(G(n, r, 0)) =

n − 1 r − 1

→
1

if p (1 + ε)pc(n, r) if p (1 − ε)pc(n, r). Su essively imp roved b y Das, T ran, Balogh, and

thers.

Let r 4 , s = 1 . Py aderkin, A.M., 2017 W.h.p. α(G1/2(n, r, s)) = α(G(n, r, s)) . Of

urse 1/2

an b e repla ed b y another fun tion. Ho w ever, the threshold is unkno wn. A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 6 / 10

SLIDE 27 Random subgraphs

f G(n, r, s) :

indep enden e numb ers fo r r > 2s + 1 Let r 2 , s = 0 . Then G(n, r, s) is Kneser's graph. Bollob

as,

Na ra y anan, A.M., 2016 Fix a real numb er ε > 0 and let r = r(n) b e a natural numb er su h that

2 r(n) = o(n1/3)

. Let pc(n, r) = ((r + 1) log n − r log r)/

n−1

r−1

.

As n → ∞,

P

α(Gp(n, r, 0)) = α(G(n, r, 0)) =

n − 1 r − 1

→
1

if p (1 + ε)pc(n, r) if p (1 − ε)pc(n, r). Su essively imp roved b y Das, T ran, Balogh, and

thers.

Let r 4 , s = 1 . Py aderkin, A.M., 2017 W.h.p. α(G1/2(n, r, s)) = α(G(n, r, s)) . Of

urse 1/2

an b e repla ed b y another fun tion. Ho w ever, the threshold is unkno wn. No

ther

ases

f

strong stabilit y a re kno wn. A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 6 / 10

SLIDE 28 Random subgraphs

f G(n, r, s) :

indep enden e numb ers fo r r 2s + 1 Remind that Theo rem (Bogoliubskiy , Gusev, Py aderkin, A.M., 20132016) Let r, s b e xed as n → ∞. If r 2s + 1 , then w.h.p.

α(G1/2(n, r, s)) = Θ (α(G(n, r, s)) log n)

. A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 7 / 10

SLIDE 29 Random subgraphs

f G(n, r, s) :

indep enden e numb ers fo r r 2s + 1 Remind that Theo rem (Bogoliubskiy , Gusev, Py aderkin, A.M., 20132016) Let r, s b e xed as n → ∞. If r 2s + 1 , then w.h.p.

α(G1/2(n, r, s)) = Θ (α(G(n, r, s)) log n)

. If r = 1 , s = 0 , then w e have already ited the mu h subtler lassi al result. A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 7 / 10

SLIDE 30 Random subgraphs

f G(n, r, s) :

indep enden e numb ers fo r r 2s + 1 Remind that Theo rem (Bogoliubskiy , Gusev, Py aderkin, A.M., 20132016) Let r, s b e xed as n → ∞. If r 2s + 1 , then w.h.p.

α(G1/2(n, r, s)) = Θ (α(G(n, r, s)) log n)

. If r = 1 , s = 0 , then w e have already ited the mu h subtler lassi al result. Theo rem Let p b e a

nstant
r

a fun tion tending to zero and b

unded

from b elo w b y a value c

n

, where c > 1 . Let d =

1 1−p

. Then w.h.p. α(Gp(n, 1, 0)) ∼ 2 logd(np). A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 7 / 10

SLIDE 31 Random subgraphs

f G(n, r, s) :

indep enden e numb ers fo r r 2s + 1 Remind that Theo rem (Bogoliubskiy , Gusev, Py aderkin, A.M., 20132016) Let r, s b e xed as n → ∞. If r 2s + 1 , then w.h.p.

α(G1/2(n, r, s)) = Θ (α(G(n, r, s)) log n)

. If r = 1 , s = 0 , then w e have already ited the mu h subtler lassi al result. Theo rem Let p b e a

nstant
r

a fun tion tending to zero and b

unded

from b elo w b y a value c

n

, where c > 1 . Let d =

1 1−p

. Then w.h.p. α(Gp(n, 1, 0)) ∼ 2 logd(np). There a re

nly

t w

mo

re ases where the Θ notation is repla ed b y the ∼

ne.

A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 7 / 10

SLIDE 32 Random subgraphs

f G(n, r, s) :

indep enden e numb ers fo r r 2s + 1 Remind that Theo rem (Bogoliubskiy , Gusev, Py aderkin, A.M., 20132016) Let r, s b e xed as n → ∞. If r 2s + 1 , then w.h.p.

α(G1/2(n, r, s)) = Θ (α(G(n, r, s)) log n)

. If r = 1 , s = 0 , then w e have already ited the mu h subtler lassi al result. Theo rem Let p b e a

nstant
r

a fun tion tending to zero and b

unded

from b elo w b y a value c

n

, where c > 1 . Let d =

1 1−p

. Then w.h.p. α(Gp(n, 1, 0)) ∼ 2 logd(np). There a re

nly

t w

mo

re ases where the Θ notation is repla ed b y the ∼

ne.

Theo rem (Py aderkin, 2016) W.h.p. α(G1/2(n, 3, 1)) ∼ 2α(G(n, 3, 1)) log2 n . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 7 / 10

SLIDE 33 Random subgraphs

f G(n, r, s) :

indep enden e numb ers fo r r 2s + 1 Remind that Theo rem (Bogoliubskiy , Gusev, Py aderkin, A.M., 20132016) Let r, s b e xed as n → ∞. If r 2s + 1 , then w.h.p.

α(G1/2(n, r, s)) = Θ (α(G(n, r, s)) log n)

. If r = 1 , s = 0 , then w e have already ited the mu h subtler lassi al result. Theo rem Let p b e a

nstant
r

a fun tion tending to zero and b

unded

from b elo w b y a value c

n

, where c > 1 . Let d =

1 1−p

. Then w.h.p. α(Gp(n, 1, 0)) ∼ 2 logd(np). There a re

nly

t w

mo

re ases where the Θ notation is repla ed b y the ∼

ne.

Theo rem (Py aderkin, 2016) W.h.p. α(G1/2(n, 3, 1)) ∼ 2α(G(n, 3, 1)) log2 n . (Kiselev, Derevy ank

,

2017) W.h.p. α(G1/2(n, 2, 1)) ∼ α(G(n, 2, 1)) log2 n . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 7 / 10

SLIDE 34 Random subgraphs

f G(n, r, s) :

hromati numb ers Let us skip rather umb ersome ases

f

a rbitra ry r, s and

n entrate
n

Kneser's graphs (r > 1 , s = 0 ). A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 8 / 10

SLIDE 35 Random subgraphs

f G(n, r, s) :

hromati numb ers Let us skip rather umb ersome ases

f

a rbitra ry r, s and

n entrate
n

Kneser's graphs (r > 1 , s = 0 ). Lov

asz,

1978: if r n/2 , then χ(G(n, r, 0)) = n − 2r + 2 . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 8 / 10

SLIDE 36 Random subgraphs

f G(n, r, s) :

hromati numb ers Let us skip rather umb ersome ases

f

a rbitra ry r, s and

n entrate
n

Kneser's graphs (r > 1 , s = 0 ). Lov

asz,

1978: if r n/2 , then χ(G(n, r, 0)) = n − 2r + 2 . V ery simply the hromati numb er

f G(n, r, 0)

is not so stable as the indep enden e numb er: w.h.p. even χ(G1/2(n, r, 0)) < n − 2r + 2 . Ho w ever A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 8 / 10

SLIDE 37 Random subgraphs

f G(n, r, s) :

hromati numb ers Let us skip rather umb ersome ases

f

a rbitra ry r, s and

n entrate
n

Kneser's graphs (r > 1 , s = 0 ). Lov

asz,

1978: if r n/2 , then χ(G(n, r, 0)) = n − 2r + 2 . V ery simply the hromati numb er

f G(n, r, 0)

is not so stable as the indep enden e numb er: w.h.p. even χ(G1/2(n, r, 0)) < n − 2r + 2 . Ho w ever Theo rem (Kupavskii, 2016) F

r

many dierent n, r, p , w.h.p. χ(Gp(n, r, 0)) ∼ n − 2r + 2. A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 8 / 10

SLIDE 38 Random subgraphs

f G(n, r, s) :

hromati numb ers Let us skip rather umb ersome ases

f

a rbitra ry r, s and

n entrate
n

Kneser's graphs (r > 1 , s = 0 ). Lov

asz,

1978: if r n/2 , then χ(G(n, r, 0)) = n − 2r + 2 . V ery simply the hromati numb er

f G(n, r, 0)

is not so stable as the indep enden e numb er: w.h.p. even χ(G1/2(n, r, 0)) < n − 2r + 2 . Ho w ever Theo rem (Kupavskii, 2016) F

r

many dierent n, r, p , w.h.p. χ(Gp(n, r, 0)) ∼ n − 2r + 2. F

r

example, if g(n) is any gro wing fun tion and r is a rbitra ry in the range b et w een 2 and n

2 − g(n)

, then fo r any xed p , χ(Gp(n, r, 0)) ∼ n − 2r + 2. A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 8 / 10

SLIDE 39 Random subgraphs

f G(n, r, s) :

hromati numb ers Let us skip rather umb ersome ases

f

a rbitra ry r, s and

n entrate
n

Kneser's graphs (r > 1 , s = 0 ). Lov

asz,

1978: if r n/2 , then χ(G(n, r, 0)) = n − 2r + 2 . V ery simply the hromati numb er

f G(n, r, 0)

is not so stable as the indep enden e numb er: w.h.p. even χ(G1/2(n, r, 0)) < n − 2r + 2 . Ho w ever Theo rem (Kupavskii, 2016) F

r

many dierent n, r, p , w.h.p. χ(Gp(n, r, 0)) ∼ n − 2r + 2. F

r

example, if g(n) is any gro wing fun tion and r is a rbitra ry in the range b et w een 2 and n

2 − g(n)

, then fo r any xed p , χ(Gp(n, r, 0)) ∼ n − 2r + 2. Many imp rovements b y Kupavskii and b y Alishahi and Hajiab

lhassan.

A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 8 / 10

SLIDE 40 Random subgraphs

f G(n, r, s) :

hromati numb ers A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 9 / 10

SLIDE 41 Random subgraphs

f G(n, r, s) :

hromati numb ers Theo rem (Kiselev, Kupavskii, 2019+) If r 3 , then w.h.p.

n − c1

2r−2

log2 n χ(G1/2(n, r, 0)) n − c2

2r−2

log2 n.

A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 9 / 10

SLIDE 42 Random subgraphs

f G(n, r, s) :

hromati numb ers Theo rem (Kiselev, Kupavskii, 2019+) If r 3 , then w.h.p.

n − c1

2r−2

log2 n χ(G1/2(n, r, 0)) n − c2

2r−2

log2 n.

If r = 2 , then w.h.p.

n − c1

2

log2 n · log2 log2 n χ(G1/2(n, r, 0)) n − c2

2r−2

log2 n · log2 log2 n.

A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 9 / 10

SLIDE 43 A general result A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 10 / 10

SLIDE 44 A general result Theo rem (A.M., 2017) Let Gn = (Vn, En) , n ∈ N, b e a sequen e

f

graphs. Let Nn = |Vn|, αn = α(Gn) . Let γn b e the maximum numb er

f

verti es

f Gn

that a re non-adja ent to b

th

verti es

f

a given edge. Assume that the quantities Nn, αn, γn a re monotone in reasing to innit y and there exists a fun tion βn su h that 1

βn > γn

and βn = o(αn) ; 2

log2 Nn = o

αn

βn

;

3

log2 Nn = o (βn − γn)

. Then w.h.p. α(Gn, 1/2) ∼ α(Gn) . A. Raigo ro dskii (MIPT) 2019 T ehran, Iran 10 / 10