Ramsey properties of random graphs Rajko Nenadov Monash University - - PowerPoint PPT Presentation

Ramsey properties of random graphs

Rajko Nenadov

Monash University 29th August 2016

Introduction A graph G is Ramsey for H

G → H

if every red/blue colouring of the edges of G contains a monochromatic copy of H

Introduction A graph G is Ramsey for H

G → H

if every red/blue colouring of the edges of G contains a monochromatic copy of H

K6 → K3

Introduction A graph G is Ramsey for H

G → H

if every red/blue colouring of the edges of G contains a monochromatic copy of H

K6 → K3

Introduction A graph G is Ramsey for H

G → H

if every red/blue colouring of the edges of G contains a monochromatic copy of H

K6 → K3

Introduction A graph G is Ramsey for H

G → H

if every red/blue colouring of the edges of G contains a monochromatic copy of H

K6 → K3

Introduction A graph G is Ramsey for H

G → H

if every red/blue colouring of the edges of G contains a monochromatic copy of H

K6 → K3

Introduction A graph G is Ramsey for H

G → H

if every red/blue colouring of the edges of G contains a monochromatic copy of H Ramsey (1930) For every graph H there exists (sufficiently large) n ∈ N such that

Kn → H

Introduction A graph G is Ramsey for H

G → H

if every red/blue colouring of the edges of G contains a monochromatic copy of H Binomial random graph G(n, p)

• n vertices
• each edge present with

probability p

Introduction A graph G is Ramsey for H

G → H

if every red/blue colouring of the edges of G contains a monochromatic copy of H Binomial random graph G(n, p)

• n vertices
• each edge present with

probability p Given a graph H and p = p(n) ∈ [0, 1], determine

Pr[G(n, p) → H]

Behaviour of Pr[G(n, p) → H] Given a graph H and p = p(n) ∈ [0, 1], determine

Pr[G(n, p) → H]

Behaviour of Pr[G(n, p) → H] Given a graph H and p = p(n) ∈ [0, 1], determine

Pr[G(n, p) → H]

• “Being Ramsey for H” is a monotone property

(preserved under edge addition)

• Bollob´

as-Thomason (’87): every non-trivial monotone property P has a threshold function p∗(P) lim

n→∞ Pr[G(n, p) ∈ P] =

( 0, if p/p∗(P) → 0 1, if p/p∗(P) → ∞

Behaviour of Pr[G(n, p) → H] Given a graph H and p = p(n) ∈ [0, 1], determine

Pr[G(n, p) → H]

• “Being Ramsey for H” is a monotone property

(preserved under edge addition)

• Bollob´

as-Thomason (’87): every non-trivial monotone property P has a threshold function p∗(P) lim

n→∞ Pr[G(n, p) ∈ P] =

( 0, if p/p∗(P) → 0 1, if p/p∗(P) → ∞ Goal: find a threshold pH for the property “being Ramsey for H”

Behaviour of Pr[G(n, p) ! H] Given a graph H and p = p(n) 2 [0, 1], determine

Pr[G(n, p) ! H]

• “Being Ramsey for H” is a monotone property

(preserved under edge addition)

• Bollob´

as-Thomason (’87): every non-trivial monotone property P has a threshold function p∗(P) lim

n→∞ Pr[G(n, p) 2 P] =

( 0, if p ⌧ p∗ 1, if p p∗ Goal: find a threshold pH for the property “being Ramsey for H”

Warm-up Given a graph H, determine pH(n) such that lim

n→∞ Pr[G(n, p) ! H] =

( 0, if p ⌧ pH 1, if p pH

Warm-up Given a graph H, determine pH(n) such that lim

n→∞ Pr[G(n, p) ! H] =

( 0, if p ⌧ pH 1, if p pH Let H = K3

Warm-up Given a graph H, determine pH(n) such that lim

n→∞ Pr[G(n, p) ! H] =

( 0, if p ⌧ pH 1, if p pH Let H = K3

• p = n−6/(6

2)+ε

lim

n→∞ Pr[G(n, p) ! K3] =

Warm-up Given a graph H, determine pH(n) such that lim

n→∞ Pr[G(n, p) ! H] =

( 0, if p ⌧ pH 1, if p pH Let H = K3

• p = n−6/(6

2)+ε

lim

n→∞ Pr[G(n, p) ! K3] = 1

Explanation: G(n, p) contains K6 with high probability

Warm-up Given a graph H, determine pH(n) such that lim

n→∞ Pr[G(n, p) ! H] =

( 0, if p ⌧ pH 1, if p pH Let H = K3

• p = n−6/(6

2)+ε

lim

n→∞ Pr[G(n, p) ! K3] = 1

Explanation: G(n, p) contains K6 with high probability

• p = n−1−ε

(n−1 < n−4/5) lim

n→∞ Pr[G(n, p) ! K3] =

Warm-up Given a graph H, determine pH(n) such that lim

n→∞ Pr[G(n, p) ! H] =

( 0, if p ⌧ pH 1, if p pH Let H = K3

• p = n−6/(6

2)+ε

lim

n→∞ Pr[G(n, p) ! K3] = 1

Explanation: G(n, p) contains K6 with high probability

• p = n−1−ε

(n−1 < n−4/5) lim

n→∞ Pr[G(n, p) ! K3] = 0

Explanation: G(n, p) does not contain K3

Warm-up Given a graph H, determine pH(n) such that lim

n→∞ Pr[G(n, p) ! H] =

( 0, if p ⌧ pH 1, if p pH Let H = K3

• p = n−6/(6

2)+ε

lim

n→∞ Pr[G(n, p) ! K3] = 1

Explanation: G(n, p) contains K6 with high probability

• p = n−4/5−ε

(n−1 < n−4/5) lim

n→∞ Pr[G(n, p) ! K3] =

Warm-up Given a graph H, determine pH(n) such that lim

n→∞ Pr[G(n, p) ! H] =

( 0, if p ⌧ pH 1, if p pH Let H = K3

• p = n−6/(6

2)+ε

lim

n→∞ Pr[G(n, p) ! K3] = 1

Explanation: G(n, p) contains K6 with high probability

• p = n−4/5−ε

(n−1 < n−4/5) lim

n→∞ Pr[G(n, p) ! K3] = 0

Explanation: G(n, p) does not contain

Warm-up Given a graph H, determine pH(n) such that lim

n→∞ Pr[G(n, p) ! H] =

( 0, if p ⌧ pH 1, if p pH Let H = K3

• p = n−6/(6

2)+ε

lim

n→∞ Pr[G(n, p) ! K3] = 1

Explanation: G(n, p) contains K6 with high probability

• p = n−5/7−ε

(n−4/5 < n−5/7) lim

n→∞ Pr[G(n, p) ! K3] =

Warm-up Given a graph H, determine pH(n) such that lim

n→∞ Pr[G(n, p) ! H] =

( 0, if p ⌧ pH 1, if p pH Let H = K3

• p = n−6/(6

2)+ε

lim

n→∞ Pr[G(n, p) ! K3] = 1

Explanation: G(n, p) contains K6 with high probability

• p = n−5/7−ε

(n−4/5 < n−5/7) lim

n→∞ Pr[G(n, p) ! K3] = 0

Explanation: G(n, p) does not contain

Warm-up Given a graph H, determine pH(n) such that lim

n→∞ Pr[G(n, p) ! H] =

( 0, if p ⌧ pH 1, if p pH Let H = K3

• p = n−6/(6

2)+ε

lim

n→∞ Pr[G(n, p) ! K3] = 1

Explanation: G(n, p) contains K6 with high probability

• p = n−2/3−ε

(n−5/7 < n−2/3) lim

n→∞ Pr[G(n, p) ! K3] =

Warm-up Given a graph H, determine pH(n) such that lim

n→∞ Pr[G(n, p) ! H] =

( 0, if p ⌧ pH 1, if p pH Let H = K3

• p = n−6/(6

2)+ε

lim

n→∞ Pr[G(n, p) ! K3] = 1

Explanation: G(n, p) contains K6 with high probability

• p = n−2/3−ε

(n−5/7 < n−2/3) lim

n→∞ Pr[G(n, p) ! K3] = 0

Explanation: G(n, p) does not contain

Warm-up Given a graph H, determine pH(n) such that lim

n→∞ Pr[G(n, p) ! H] =

( 0, if p ⌧ pH 1, if p pH Let H = K3

• p = n−6/(6

2)+ε

lim

n→∞ Pr[G(n, p) ! K3] = 1

Explanation: G(n, p) contains K6 with high probability

• p = n−1/2−ε

(n−2/3 < n−1/2) lim

n→∞ Pr[G(n, p) ! K3] = 0

Explanation: whiteboard

Threshold for G(n, p) → K3 Frankl-R¨

• dl (’86),

Luczak-Ruci´ nski-Voigt (’92’) There exist constants c, C > 0 such that lim

n→∞ Pr[G(n, p) → K3] =

( 0, if p ≤ cn−1/2 1, if p ≥ Cn−1/2

Threshold for G(n, p) → H R¨

• dl-Ruci´

nski (’93–’95) For every graph H (which contains a cycle) there exist constants c, C > 0 such that lim

n→∞ Pr[G(n, p) → H] =

( 0, if p ≤ cn−β(H) 1, if p ≥ Cn−β(H)

Threshold for G(n, p) → H R¨

• dl-Ruci´

nski (’93–’95) For every graph H (which contains a cycle) there exist constants c, C > 0 such that lim

n→∞ Pr[G(n, p) → H] =

( 0, if p ≤ cn−β(H) 1, if p ≥ Cn−β(H) Intuition: β(H) is chosen such that

• p ≤ cn−β(H)

→ most of the edges do not belong to a copy of H

• p ≥ Cn−β(H)

→ each edge belongs to many copies of H

Threshold for G(n, p) → H R¨

• dl-Ruci´

nski (’93–’95) For every graph H (which contains a cycle) there exist constants c, C > 0 such that lim

n→∞ Pr[G(n, p) → H] =

( 0, if p ≤ cn−β(H) 1, if p ≥ Cn−β(H) Intuition: β(H) is chosen such that

• p ≤ cn−β(H)

→ most of the edges do not belong to a copy of H

• p ≥ Cn−β(H)

→ each edge belongs to many copies of H N.-Steger (2015) – a ‘short’ proof

lim

n→∞ Pr[G(n, p) → K3] = 1 for p ≥ Cn−1/2

lim

n→∞ Pr[G(n, p) → K3] = 1 for p ≥ Cn−1/2

• r

‘A short introduction to hypergraph containers’

Hypergraph containers Balogh–Morris–Samotij and Saxton-Thomason (2015) ∀δ > 0 ∃K > 0: for every n ∈ N there exists a collection C of graphs on n vertices and a function f : 2E(Kn) → C such that (a) each C ∈ C contains at most δn3 triangles, (b) for every K3-free graph H there exists S ⊆ E(Kn) such that e(S) ≤ Kn3/2 and S ⊆ H ⊆ f(S)

• C contains all triangle-free graphs (containers)
• container of a graph H is generated by its small subgraph

Proof

G(n, p) → K3 for p ≥ Cn−1/2

Let G ⊆ Kn be a graph on n vertices

Proof

G(n, p) → K3 for p ≥ Cn−1/2

Let G ⊆ Kn be a graph on n vertices

• Suppose there exists a colouring

E(G) = R ∪ B with no monochromatic triangle

Proof

G(n, p) → K3 for p ≥ Cn−1/2

Let G ⊆ Kn be a graph on n vertices

• Suppose there exists a colouring

E(G) = R ∪ B with no monochromatic triangle

Proof

G(n, p) → K3 for p ≥ Cn−1/2

Let G ⊆ Kn be a graph on n vertices

• Suppose there exists a colouring

E(G) = R ∪ B with no monochromatic triangle (both R and B are triangle-free)

Proof

G(n, p) → K3 for p ≥ Cn−1/2

Let G ⊆ Kn be a graph on n vertices

• Suppose there exists a colouring

E(G) = R ∪ B with no monochromatic triangle (both R and B are triangle-free)

• Container theorem – there exist

small subgraphs SR, SB ⊆ E(Kn) such that SR ⊆ R ⊆ f(SR) SB ⊆ B ⊆ f(SB) Proof

G(n, p) → K3 for p ≥ Cn−1/2

Let G ⊆ Kn be a graph on n vertices

• Suppose there exists a colouring

E(G) = R ∪ B with no monochromatic triangle (both R and B are triangle-free)

• Container theorem – there exist

small subgraphs SR, SB ⊆ E(Kn) such that SR ⊆ R ⊆ f(SR) SB ⊆ B ⊆ f(SB) Recall: e(SR) < Kn3/2, f(SR) contains at most δn3 triangles

Proof

G(n, p) → K3 for p ≥ Cn−1/2

Let G ⊆ Kn be a graph on n vertices

• Suppose there exists a colouring

E(G) = R ∪ B with no monochromatic triangle (both R and B are triangle-free)

• Container theorem – there exist

small subgraphs SR, SB ⊆ E(Kn) such that SR ⊆ R ⊆ f(SR) SB ⊆ B ⊆ f(SB) Recall: e(SR) < Kn3/2, f(SR) contains at most δn3 triangles

Proof

G(n, p) → K3 for p ≥ Cn−1/2

Let G ⊆ Kn be a graph on n vertices

• Suppose there exists a colouring

E(G) = R ∪ B with no monochromatic triangle (both R and B are triangle-free)

• Container theorem – there exist

small subgraphs SR, SB ⊆ E(Kn) such that SR ⊆ R ⊆ f(SR) SB ⊆ B ⊆ f(SB) Recall: e(SR) < Kn3/2, f(SR) contains at most δn3 triangles

Proof

G(n, p) → K3 for p ≥ Cn−1/2

Let G ⊆ Kn be a graph on n vertices

• Suppose there exists a colouring

E(G) = R ∪ B with no monochromatic triangle (both R and B are triangle-free)

• Container theorem – there exist

small subgraphs SR, SB ⊆ E(Kn) such that SR ⊆ R ⊆ f(SR) SB ⊆ B ⊆ f(SB) Recall: e(SR) < Kn3/2, f(SR) contains at most δn3 triangles

Proof

G(n, p) → K3 for p ≥ Cn−1/2

Let G ⊆ Kn be a graph on n vertices

• Suppose there exists a colouring

E(G) = R ∪ B with no monochromatic triangle (both R and B are triangle-free)

• Container theorem – there exist

small subgraphs SR, SB ⊆ E(Kn) such that SR ⊆ R ⊆ f(SR) SB ⊆ B ⊆ f(SB) Recall: e(SR) < Kn3/2, f(SR) contains at most δn3 triangles

Proof

G(n, p) → K3 for p ≥ Cn−1/2

Let G ⊆ Kn be a graph on n vertices

• Suppose there exists a colouring

E(G) = R ∪ B with no monochromatic triangle (both R and B are triangle-free)

• Container theorem – there exist

small subgraphs SR, SB ⊆ E(Kn) such that SR ⊆ R ⊆ f(SR) SB ⊆ B ⊆ f(SB) Recall: e(SR) < Kn3/2, f(SR) contains at most δn3 triangles

• L = Kn \ (f(SR) ∪ f(SB))

Proof

G(n, p) → K3 for p ≥ Cn−1/2

Let G ⊆ Kn be a graph on n vertices

• Suppose there exists a colouring

E(G) = R ∪ B with no monochromatic triangle (both R and B are triangle-free)

• Container theorem – there exist

small subgraphs SR, SB ⊆ E(Kn) such that SR ⊆ R ⊆ f(SR) SB ⊆ B ⊆ f(SB) Recall: e(SR) < Kn3/2, f(SR) contains at most δn3 triangles

• L = Kn \ (f(SR) ∪ f(SB))

Proof

G(n, p) → K3 for p ≥ Cn−1/2

Let G ⊆ Kn be a graph on n vertices

• Suppose there exists a colouring

E(G) = R ∪ B with no monochromatic triangle (both R and B are triangle-free)

• Container theorem – there exist

small subgraphs SR, SB ⊆ E(Kn) such that SR ⊆ R ⊆ f(SR) SB ⊆ B ⊆ f(SB) Recall: e(SR) < Kn3/2, f(SR) contains at most δn3 triangles

• L = Kn \ (f(SR) ∪ f(SB))
• Crucial observations:

L ∩ G = ∅, e(L) ≥ αn2

Threshold for G(n, p) → H R¨

• dl-Ruci´

nski (’93–’95) For every graph H (which contains a cycle) there exist constants c, C > 0 such that lim

n→∞ Pr[G(n, p) → H] =

( 0, if p ≤ cn−β(H) 1, if p ≥ Cn−β(H)

Other approaches and generalisations Proof (1-statement):

• Multiple exposure
• Sparse Regularity Lemma +

K LR Generalisations:

• Hypergraphs
• Asymmetric Ramsey properties
• Sharp threshold

Other approaches and generalisations Proof (1-statement):

• Multiple exposure

Generalisations:

• Hypergraphs

Other approaches and generalisations Proof (1-statement):

• Multiple exposure

Generalisations:

• Hypergraphs
• Original proof of R¨
• dl-Ruci´

nski

• Basic idea:
• G(n, p) = G(n, p1) ∪ G(n, p2)

where p1 = αp2

• Show that any colouring of G(n, p1) either contains a mono.

H or heaps of well-distributed mono. H − e

• There are 2n2p1 colourings
• Show that with probability e−Ω(n2p2) any extension to a

colouring to G(n, p2) gives a mono. H

Other approaches and generalisations Proof (1-statement):

• Multiple exposure

Generalisations:

• Hypergraphs
• Original proof of R¨
• dl-Ruci´

nski

• Basic idea:
• G(n, p) = G(n, p1) ∪ G(n, p2)

where p1 = αp2

• Show that any colouring of G(n, p1) either contains a mono.

H or heaps of well-distributed mono. H − e

• There are 2n2p1 colourings
• Show that with probability e−Ω(n2p2) any extension to a

colouring to G(n, p2) gives a mono. H

Other approaches and generalisations Proof (1-statement):

• Multiple exposure

Generalisations:

• Hypergraphs
• Original proof of R¨
• dl-Ruci´

nski

• Basic idea:
• G(n, p) = G(n, p1) ∪ G(n, p2)

where p1 = αp2

• Show that any colouring of G(n, p1) either contains a mono.

H or heaps of well-distributed mono. H − e

• There are 2n2p1 colourings
• Show that with probability e−Ω(n2p2) any extension to a

colouring to G(n, p2) gives a mono. H

Other approaches and generalisations Proof (1-statement):

• Multiple exposure
• Sparse Regularity Lemma +

K LR Generalisations:

• Hypergraphs
• Asymmetric Ramsey properties
• Sharp threshold

Other approaches and generalisations Proof (1-statement):

• Multiple exposure
• Sparse Regularity Lemma +

K LR Generalisations:

• Hypergraphs
• Asymmetric Ramsey properties
• Sharp threshold
• Folklore – dates back to Chv´

atal-R¨

• dl-Szemer´

edi-Trotter (’83)

• K

LR conjecture (Kohayakawa, Luczak, R¨

• dl) posed as a way

to tackle Turan’s theorem for random graphs

• Solved using di↵erent techniques by Conlon–Gowers and

Schacht (2016, Annals of Mathematics)

• Many partial results until finally settled
• Balogh-Morris-Samotij and Saxton-Thomason

(2015, containers)

• Conlon, Gowers, Samotij, Schacht

(2014, weaker in one sense/stronger in the other)

Other approaches and generalisations Proof (1-statement):

• Multiple exposure
• Sparse Regularity Lemma +

K LR Generalisations:

• Hypergraphs
• Asymmetric Ramsey properties
• Sharp threshold
• Folklore – dates back to Chv´

atal-R¨

• dl-Szemer´

edi-Trotter (’83)

• K

LR conjecture (Kohayakawa, Luczak, R¨

• dl) posed as a way

to tackle Turan’s theorem for random graphs

• Solved using di↵erent techniques by Conlon–Gowers and

Schacht (2016, Annals of Mathematics)

• Many partial results until finally settled
• Balogh-Morris-Samotij and Saxton-Thomason

(2015, containers)

• Conlon, Gowers, Samotij, Schacht

(2014, weaker in one sense/stronger in the other)

Other approaches and generalisations Proof (1-statement):

• Multiple exposure
• Sparse Regularity Lemma +

K LR Generalisations:

• Hypergraphs
• Asymmetric Ramsey properties
• Sharp threshold
• Folklore – dates back to Chv´

atal-R¨

• dl-Szemer´

edi-Trotter (’83)

• K

LR conjecture (Kohayakawa, Luczak, R¨

• dl) posed as a way

to tackle Turan’s theorem for random graphs

• Solved using di↵erent techniques by Conlon–Gowers and

Schacht (2016, Annals of Mathematics)

• Many partial results until finally settled
• Balogh-Morris-Samotij and Saxton-Thomason

(2015, containers)

• Conlon, Gowers, Samotij, Schacht

(2014, weaker in one sense/stronger in the other)

Other approaches and generalisations Proof (1-statement):

• Multiple exposure
• Sparse Regularity Lemma +

K LR Generalisations:

• Hypergraphs
• Asymmetric Ramsey properties
• Sharp threshold
• Folklore – dates back to Chv´

atal-R¨

• dl-Szemer´

edi-Trotter (’83)

• K

LR conjecture (Kohayakawa, Luczak, R¨

• dl) posed as a way

to tackle Turan’s theorem for random graphs

• Solved using di↵erent techniques by Conlon–Gowers and

Schacht (2016, Annals of Mathematics)

• Many partial results until finally settled
• Balogh-Morris-Samotij and Saxton-Thomason

(2015, containers)

• Conlon, Gowers, Samotij, Schacht

(2014, weaker in one sense/stronger in the other)

Other approaches and generalisations Proof (1-statement):

• Multiple exposure
• Sparse Regularity Lemma +

K LR Generalisations:

• Hypergraphs
• Asymmetric Ramsey properties
• Sharp threshold

Hypergraphs Friedgut-R¨

• dl-Schacht (’10) and Conlon-Gowers (’16)

For every k-hypergraph H there exists C > 0 such that if p ≥ Cn−β(H) then lim

n→∞ Pr[G(k)(n, p) → H] = 1

β(H) is chosen such that

• p ≤ cn−β(H) (for some small constant c > 0)

→ most of the hyperedges do not belong to a copy of H

• p ≥ Cn−β(H)

→ each hyperedge belongs to many copies of H

Hypergraphs Friedgut-R¨

• dl-Schacht (’10) and Conlon-Gowers (’16)

For every k-hypergraph H there exists C > 0 such that if p ≥ Cn−β(H) then lim

n→∞ Pr[G(k)(n, p) → H] = 1

Corresponding statement for p < cn−β(H)

• Partial results (hypergraph cliques, etc.)

N.-Person-Steger-ˇ Skori´ c (’16+), PhD thesis of Thomas

Hypergraphs Friedgut-R¨

• dl-Schacht (’10) and Conlon-Gowers (’16)

For every k-hypergraph H there exists C > 0 such that if p ≥ Cn−β(H) then lim

n→∞ Pr[G(k)(n, p) → H] = 1

Corresponding statement for p < cn−β(H)

• Partial results (hypergraph cliques, etc.)

N.-Person-Steger-ˇ Skori´ c (’16+), PhD thesis of Thomas

• Connection to asymmetric Ramsey properties

N.-Person-Steger-ˇ Skori´ c (’16+)

Other approaches and generalisations Proof (1-statement):

• Multiple exposure
• Sparse Regularity Lemma +

K LR Generalisations:

• Hypergraphs
• Asymmetric Ramsey properties
• Sharp threshold

Other approaches and generalisations Proof (1-statement):

• Multiple exposure
• Sparse Regularity Lemma +

K LR Generalisations:

• Hypergraphs
• Asymmetric Ramsey properties
• Sharp threshold
• Instead of avoiding H in both colours, avoid H1 in red and

H2 in blue

• If H2 is sparser than H1, then the threshold is smaller than for

G(n, p) → H1

• Kohayakawa-Kreuter (’96) – cycles
• Marciniszyn-Skokan-Sp¨
• hel-Steger (’08) – cliques
• 1-statement
• Sparse Regularity Lemma + K

LR

• Multiple exposure: Kohayakawa-Schacht-Sp¨
• hel (’14)
• Containers: N.-Person-Steger-ˇ

Skori´ c (’16+) (gives the hypergraph version)

Other approaches and generalisations Proof (1-statement):

• Multiple exposure
• Sparse Regularity Lemma +

K LR Generalisations:

• Hypergraphs
• Asymmetric Ramsey properties
• Sharp threshold
• Instead of avoiding H in both colours, avoid H1 in red and

H2 in blue

• If H2 is sparser than H1, then the threshold is smaller than for

G(n, p) → H1

• Kohayakawa-Kreuter (’96) – cycles
• Marciniszyn-Skokan-Sp¨
• hel-Steger (’08) – cliques
• 1-statement
• Sparse Regularity Lemma + K

LR

• Multiple exposure: Kohayakawa-Schacht-Sp¨
• hel (’14)
• Containers: N.-Person-Steger-ˇ

Skori´ c (’16+) (gives the hypergraph version)

Other approaches and generalisations Proof (1-statement):

• Multiple exposure
• Sparse Regularity Lemma +

K LR Generalisations:

• Hypergraphs
• Asymmetric Ramsey properties
• Sharp threshold
• Instead of avoiding H in both colours, avoid H1 in red and

H2 in blue

• If H2 is sparser than H1, then the threshold is smaller than for

G(n, p) → H1

• Kohayakawa-Kreuter (’96) – cycles
• Marciniszyn-Skokan-Sp¨
• hel-Steger (’08) – cliques
• 1-statement
• Sparse Regularity Lemma + K

LR

• Multiple exposure: Kohayakawa-Schacht-Sp¨
• hel (’14)
• Containers: N.-Person-Steger-ˇ

Skori´ c (’16+) (gives the hypergraph version)

Other approaches and generalisations Proof (1-statement):

• Multiple exposure
• Sparse Regularity Lemma +

K LR Generalisations:

• Hypergraphs
• Asymmetric Ramsey properties
• Sharp threshold
• Instead of avoiding H in both colours, avoid H1 in red and

H2 in blue

• If H2 is sparser than H1, then the threshold is smaller than for

G(n, p) → H1

• Kohayakawa-Kreuter (’96) – cycles
• Marciniszyn-Skokan-Sp¨
• hel-Steger (’08) – cliques
• 1-statement
• Sparse Regularity Lemma + K

LR

• Multiple exposure: Kohayakawa-Schacht-Sp¨
• hel (’14)
• Containers: N.-Person-Steger-ˇ

Skori´ c (’16+) (gives the hypergraph version)

Other approaches and generalisations Proof (1-statement):

• Multiple exposure
• Sparse Regularity Lemma +

K LR Generalisations:

• Hypergraphs
• Asymmetric Ramsey properties
• Sharp threshold

Sharp threshold R¨

• dl-Ruci´

nski For every graph H (which contains a cycle) there exist constants c, C > 0 such that lim

n→∞ Pr[G(n, p) → H] =

( 0, if p ≤ cn−β(H) 1, if p ≥ Cn−β(H) How close are c and C?

Sharp threshold R¨

• dl-Ruci´

nski For every graph H (which contains a cycle) there exist constants c, C > 0 such that lim

n→∞ Pr[G(n, p) → H] =

( 0, if p ≤ cn−β(H) 1, if p ≥ Cn−β(H) How close are c and C? Friedgut-R¨

• dl-Schacht-Tetali (’04)

There exist constants c0, c1 > 0 and a function c(n) with c0 < c(n) < c1 such that lim

n→∞ Pr[G(n, p) → K3] =

( 0, if p ≤ (1 − ε)c(n)n−1/2 1, if p ≥ (1 + ε)c(n)n−1/2

Sharp threshold Friedgut-R¨

• dl-Schacht-Tetali (’04)

There exist constants c0, c1 > 0 and a function c(n) with c0 < c(n) < c1 such that lim

n→∞ Pr[G(n, p) → K3] =

( 0, if p ≤ (1 − ε)c(n)n−1/2 1, if p ≥ (1 + ε)c(n)n−1/2

• Friedgut-H´

an-Person-Schacht (’16) – bipartite graphs

Sharp threshold Friedgut-R¨

• dl-Schacht-Tetali (’04)

There exist constants c0, c1 > 0 and a function c(n) with c0 < c(n) < c1 such that lim

n→∞ Pr[G(n, p) → K3] =

( 0, if p ≤ (1 − ε)c(n)n−1/2 1, if p ≥ (1 + ε)c(n)n−1/2

• Friedgut-H´

an-Person-Schacht (’16) – bipartite graphs

• Schacht-Schulenburg (’16+) – bipartite graphs + an edge

Sharp threshold Friedgut-R¨

• dl-Schacht-Tetali (’04)

There exist constants c0, c1 > 0 and a function c(n) with c0 < c(n) < c1 such that lim

n→∞ Pr[G(n, p) → K3] =

( 0, if p ≤ (1 − ε)c(n)n−1/2 1, if p ≥ (1 + ε)c(n)n−1/2

• Friedgut-H´

an-Person-Schacht (’16) – bipartite graphs

• Schacht-Schulenburg (’16+) – bipartite graphs + an edge

Open problems:

• Replace c(n) by a constant (difficult)

Sharp threshold Friedgut-R¨

• dl-Schacht-Tetali (’04)

There exist constants c0, c1 > 0 and a function c(n) with c0 < c(n) < c1 such that lim

n→∞ Pr[G(n, p) → K3] =

( 0, if p ≤ (1 − ε)c(n)n−1/2 1, if p ≥ (1 + ε)c(n)n−1/2

• Friedgut-H´

an-Person-Schacht (’16) – bipartite graphs

• Schacht-Schulenburg (’16+) – bipartite graphs + an edge

Open problems:

• Replace c(n) by a constant (difficult)
• Extend to multiple colours

Sharp threshold Friedgut-R¨

• dl-Schacht-Tetali (’04)

There exist constants c0, c1 > 0 and a function c(n) with c0 < c(n) < c1 such that lim

n→∞ Pr[G(n, p) → K3] =

( 0, if p ≤ (1 − ε)c(n)n−1/2 1, if p ≥ (1 + ε)c(n)n−1/2

• Friedgut-H´

an-Person-Schacht (’16) – bipartite graphs

• Schacht-Schulenburg (’16+) – bipartite graphs + an edge

Open problems:

• Replace c(n) by a constant (difficult)
• Extend to multiple colours
• Extend to cliques

Further applications – Folkman numbers Given r, k 2 N, what is the smallest number f(r, k) for which there exists a graph G on f(r, k) vertices such that G ! (Kr)k and Kr+1 6✓ G?

Further applications – Folkman numbers Given r, k 2 N, what is the smallest number f(r, k) for which there exists a graph G on f(r, k) vertices such that G ! (Kr)k and Kr+1 6✓ G? Existence:

• Folkman (’70) – f(r, 2) (answers a question of Erd˝
• s-Hajnal)

Further applications – Folkman numbers Given r, k 2 N, what is the smallest number f(r, k) for which there exists a graph G on f(r, k) vertices such that G ! (Kr)k and Kr+1 6✓ G? Existence:

• Folkman (’70) – f(r, 2) (answers a question of Erd˝
• s-Hajnal)
• Neˇ

setˇ ril-R¨

• dl (’76) – arbitrary k

Further applications – Folkman numbers Given r, k 2 N, what is the smallest number f(r, k) for which there exists a graph G on f(r, k) vertices such that G ! (Kr)k and Kr+1 6✓ G? Existence:

• Folkman (’70) – f(r, 2) (answers a question of Erd˝
• s-Hajnal)
• Neˇ

setˇ ril-R¨

• dl (’76) – arbitrary k

Quantitative bounds:

• R¨
• dl-Ruci´

nski-Schacht (’16+) – f(k)  2kck2 (mult. exp.)

Further applications – Folkman numbers Given r, k 2 N, what is the smallest number f(r, k) for which there exists a graph G on f(r, k) vertices such that G ! (Kr)k and Kr+1 6✓ G? Existence:

• Folkman (’70) – f(r, 2) (answers a question of Erd˝
• s-Hajnal)
• Neˇ

setˇ ril-R¨

• dl (’76) – arbitrary k

Quantitative bounds:

• R¨
• dl-Ruci´

nski-Schacht (’16+) – f(k)  2kck2 (mult. exp.)

• Similar results by Conlon-Gowers (’16+) – multicolour case

Further applications – Folkman numbers Given r, k 2 N, what is the smallest number f(r, k) for which there exists a graph G on f(r, k) vertices such that G ! (Kr)k and Kr+1 6✓ G? Existence:

• Folkman (’70) – f(r, 2) (answers a question of Erd˝
• s-Hajnal)
• Neˇ

setˇ ril-R¨

• dl (’76) – arbitrary k

Quantitative bounds:

• R¨
• dl-Ruci´

nski-Schacht (’16+) – f(k)  2kck2 (mult. exp.)

• Similar results by Conlon-Gowers (’16+) – multicolour case
• R¨
• dl-Ruci´

nski-Schacht (’16+) – f(k, r)  2O(k4 log k+k3r log r) (similar to the presented proof)

Thank you!