On the size Ramsey number

On the size Ramsey number

Dennis Clemens

Monash University Melbourne March 19

21

2018

joint work with: (1) Matthew Jenssen, Yoshiharu Kohayakawa, Natasha Morrison, Guilherme O. Mota, Damian Reding, Barnaby Roberts &

(2) Meysam Miralaei, Damian Reding, Mathias Schacht, Anusch Taraz

Ramsey graphs

Ramsey graphs

Definition: A graph G is Ramsey for a graph F , if no matter how one colors

the edges of G with 2 colors, one of the color classes contains a copy of F .

Ramsey graphs

Definition: A graph G is Ramsey for a graph F , if no matter how one colors

the edges of G with 2 colors, one of the color classes contains a copy of F .

Notation: G → F to denote that G is Ramsey for F

Ramsey graphs

Definition: A graph G is Ramsey for a graph F , if no matter how one colors the edges of G with 2 colors, one of the color classes contains a copy of F . Notation: G → F to denote that G is Ramsey for F

Typical example: K

→ K

and K

6→ K

Ramsey graphs

Definition: A graph G is Ramsey for a graph F , if no matter how one colors the edges of G with 2 colors, one of the color classes contains a copy of F . Notation: G → F to denote that G is Ramsey for F

Typical example: K

→ K

and K

6→ K

Ramsey graphs

Definition: A graph G is Ramsey for a graph F , if no matter how one colors the edges of G with 2 colors, one of the color classes contains a copy of F . Notation: G → F to denote that G is Ramsey for F

Typical example: K

→ K

and K

6→ K

Theorem (Ramsey; 1930)

For every graph F , there is an integer n ∈ N such that K

→ F

Ramsey graphs

Definition: A graph G is Ramsey for a graph F , if no matter how one colors the edges of G with 2 colors, one of the color classes contains a copy of F . Notation: G → F to denote that G is Ramsey for F

Typical example: K

→ K

and K

6→ K

Theorem (Ramsey; 1930)

For every graph F , there is an integer n ∈ N such that K

→ F

Typical questions: Given a graph F . Determine

r (F ) := min {v(G) : G → F } [Ramsey number]

s(F ) := min {δ(G) : G → F and H 6→ F for all H ( G}

ˆ r (F ) := min {e(G) : G → F } [size Ramsey number]

(... and other graph parameters)

Ramsey graphs

Definition: A graph G is Ramsey for a graph F , if no matter how one colors the edges of G with 2 colors, one of the color classes contains a copy of F . Notation: G → F to denote that G is Ramsey for F

Typical example: K

→ K

and K

6→ K

Theorem (Ramsey; 1930)

For every graph F , there is an integer n ∈ N such that K

→ F

Typical questions: Given a graph F . Determine

r (F ) := min {v(G) : G → F } [Ramsey number]

s(F ) := min {δ(G) : G → F and H 6→ F for all H ( G}

ˆ r (F ) := min {e(G) : G → F } [size Ramsey number]

(... and other graph parameters)

Ramsey graphs

Definition: A graph G is Ramsey for a graph F , if no matter how one colors the edges of G with 2 colors, one of the color classes contains a copy of F . Notation: G → F to denote that G is Ramsey for F

Typical example: K

→ K

and K

6→ K

Theorem (Ramsey; 1930)

For every graph F , there is an integer n ∈ N such that K

→ F

Typical questions: Given a graph F . Determine

r (F ) := min {v(G) : G → F } [Ramsey number]

s(F ) := min {δ(G) : G → F and H 6→ F for all H ( G}

ˆ r (F ) := min {e(G) : G → F } [size Ramsey number]

(... and other graph parameters)

Ramsey graphs

Definition: A graph G is Ramsey for a graph F , if no matter how one colors the edges of G with 2 colors, one of the color classes contains a copy of F . Notation: G → F to denote that G is Ramsey for F

Typical example: K

→ K

and K

6→ K

Theorem (Ramsey; 1930)

For every graph F , there is an integer n ∈ N such that K

→ F

Typical questions: Given a graph F . Determine

r (F ) := min {v(G) : G → F } [Ramsey number]

s(F ) := min {δ(G) : G → F and H 6→ F for all H ( G}

ˆ r (F ) := min {e(G) : G → F } [size Ramsey number]

(... and other graph parameters)

Ramsey graphs

Definition: A graph G is Ramsey for a graph F , if no matter how one colors the edges of G with 2 colors, one of the color classes contains a copy of F . Notation: G → F to denote that G is Ramsey for F

Typical example: K

→ K

and K

6→ K

Theorem (Ramsey; 1930)

For every graph F , there is an integer n ∈ N such that K

→ F

Typical questions: Given a graph F . Determine

r (F ) := min {v(G) : G → F } [Ramsey number]

s(F ) := min {δ(G) : G → F and H 6→ F for all H ( G}

ˆ r (F ) := min {e(G) : G → F } [size Ramsey number]

(... and other graph parameters)

Size Ramsey number

Definition (Erd ˝os, Faudree, Rousseau, Schelp; 1978) The size Ramsey number of a graph F is defined as

ˆ r (F ) := min {e(G) : G → F } .

Observation: For every graph F we have ˆ r (F ) ≤ r (F )

2

 .

Erd ˝os et. al. (1978) asked for graphs for which equality holds and for graph sequences for which ˆ r (F ) is of smaller order than

.

Theorem (Chvátal; 1978)

ˆ r (K

) = r (K

) 2



Question (Erd ˝os; 1981) Is it true that lim

n→∞

ˆ r (P

)

n = ∞ and lim

n→∞

ˆ r (P

)

n

= 0 ?

Size Ramsey number

Definition (Erd ˝os, Faudree, Rousseau, Schelp; 1978) The size Ramsey number of a graph F is defined as

ˆ r (F ) := min {e(G) : G → F } . Observation: For every graph F we have

ˆ r (F ) ≤ r (F ) 2

 .

Erd ˝os et. al. (1978) asked for graphs for which equality holds and for graph sequences for which ˆ r (F ) is of smaller order than

.

Theorem (Chvátal; 1978)

ˆ r (K

) = r (K

) 2



Question (Erd ˝os; 1981) Is it true that lim

n→∞

ˆ r (P

)

n = ∞ and lim

n→∞

ˆ r (P

)

n

= 0 ?

Size Ramsey number

Definition (Erd ˝os, Faudree, Rousseau, Schelp; 1978) The size Ramsey number of a graph F is defined as

ˆ r (F ) := min {e(G) : G → F } . Observation: For every graph F we have

ˆ r (F ) ≤ r (F ) 2

 .

Erd ˝os et. al. (1978) asked for graphs for which equality holds

and for graph sequences for which ˆ r (F ) is of smaller order than

.

Theorem (Chvátal; 1978)

ˆ r (K

) = r (K

) 2



Question (Erd ˝os; 1981) Is it true that lim

n→∞

ˆ r (P

)

n = ∞ and lim

n→∞

ˆ r (P

)

n

= 0 ?

Size Ramsey number

Definition (Erd ˝os, Faudree, Rousseau, Schelp; 1978) The size Ramsey number of a graph F is defined as

ˆ r (F ) := min {e(G) : G → F } . Observation: For every graph F we have

ˆ r (F ) ≤ r (F ) 2

 .

Erd ˝os et. al. (1978) asked for graphs for which equality holds

and for graph sequences for which ˆ r (F ) is of smaller order than

.

Theorem (Chvátal; 1978)

ˆ r (K

) = r (K

) 2



Question (Erd ˝os; 1981) Is it true that lim

n→∞

ˆ r (P

)

n = ∞ and lim

n→∞

ˆ r (P

)

n

= 0 ?

Size Ramsey number

Definition (Erd ˝os, Faudree, Rousseau, Schelp; 1978) The size Ramsey number of a graph F is defined as

ˆ r (F ) := min {e(G) : G → F } . Observation: For every graph F we have

ˆ r (F ) ≤ r (F ) 2

 .

Erd ˝os et. al. (1978) asked for graphs for which equality holds and for graph sequences for which ˆ r (F ) is of smaller order than

.

Theorem (Chvátal; 1978)

ˆ r (K

) = r (K

) 2



Question (Erd ˝os; 1981) Is it true that lim

n→∞

ˆ r (P

)

n = ∞ and lim

n→∞

ˆ r (P

)

n

= 0 ?

Size Ramsey number

Definition (Erd ˝os, Faudree, Rousseau, Schelp; 1978) The size Ramsey number of a graph F is defined as

ˆ r (F ) := min {e(G) : G → F } . Observation: For every graph F we have

ˆ r (F ) ≤ r (F ) 2

 .

Erd ˝os et. al. (1978) asked for graphs for which equality holds and for graph sequences for which ˆ r (F ) is of smaller order than

.

Theorem (Chvátal; 1978)

ˆ r (K

) = r (K

) 2



Question (Erd ˝os; 1981) Is it true that lim

n→∞

ˆ r (P

)

n = ∞ and lim

n→∞

ˆ r (P

)

n

= 0 ?

Size Ramsey number of P _n

Question (Erd ˝os; 1981) Is it true that lim

n→∞

ˆ r (P

)

n = ∞ and lim

n→∞

ˆ r (P

) n

= 0 ?

Theorem (Beck; 1983)

There is a constant C > 0 such that for every n ∈ N it holds that ˆr(P

) < Cn . For short:

ˆ r (P

) = O(n) .

Beck’s proof uses a probabilistic construction (and a large constant C) Alon, Chung (1988) gave an explicit construction of a graph G → P

with e(G) = Θ(n)

Dudek, Prałat (2015) gave a simple (probabilistic) proof showing that ˆ r (P

) < 137n

Natural question: For the size Ramsey numbers of which graph sequences

do we have linearity in the number of vertices?

Size Ramsey number of P _n

Question (Erd ˝os; 1981) Is it true that lim

n→∞

ˆ r (P

)

n = ∞ and lim

n→∞

ˆ r (P

) n

= 0 ? Theorem (Beck; 1983)

There is a constant C > 0 such that for every n ∈ N it holds that ˆr(P

) < Cn . For short:

ˆ r (P

) = O(n) .

Beck’s proof uses a probabilistic construction (and a large constant C) Alon, Chung (1988) gave an explicit construction of a graph G → P

with e(G) = Θ(n)

Dudek, Prałat (2015) gave a simple (probabilistic) proof showing that ˆ r (P

) < 137n

Natural question: For the size Ramsey numbers of which graph sequences

do we have linearity in the number of vertices?

Size Ramsey number of P _n

Question (Erd ˝os; 1981) Is it true that lim

n→∞

ˆ r (P

)

n = ∞ and lim

n→∞

ˆ r (P

) n

= 0 ? Theorem (Beck; 1983)

There is a constant C > 0 such that for every n ∈ N it holds that ˆr(P

) < Cn . For short:

ˆ r (P

) = O(n) .

Beck’s proof uses a probabilistic construction (and a large constant C)

Alon, Chung (1988) gave an explicit construction of a graph G → P

with e(G) = Θ(n)

Dudek, Prałat (2015) gave a simple (probabilistic) proof showing that ˆ r (P

) < 137n

Natural question: For the size Ramsey numbers of which graph sequences

do we have linearity in the number of vertices?

Size Ramsey number of P _n

Question (Erd ˝os; 1981) Is it true that lim

n→∞

ˆ r (P

)

n = ∞ and lim

n→∞

ˆ r (P

) n

= 0 ? Theorem (Beck; 1983)

There is a constant C > 0 such that for every n ∈ N it holds that ˆr(P

) < Cn . For short:

ˆ r (P

) = O(n) .

Beck’s proof uses a probabilistic construction (and a large constant C) Alon, Chung (1988) gave an explicit construction of a graph G → P

with e(G) = Θ(n)

Dudek, Prałat (2015) gave a simple (probabilistic) proof showing that ˆ r (P

) < 137n

Natural question: For the size Ramsey numbers of which graph sequences

do we have linearity in the number of vertices?

Size Ramsey number of P _n

Question (Erd ˝os; 1981) Is it true that lim

n→∞

ˆ r (P

)

n = ∞ and lim

n→∞

ˆ r (P

) n

= 0 ? Theorem (Beck; 1983)

There is a constant C > 0 such that for every n ∈ N it holds that ˆr(P

) < Cn . For short:

ˆ r (P

) = O(n) .

Beck’s proof uses a probabilistic construction (and a large constant C) Alon, Chung (1988) gave an explicit construction of a graph G → P

with e(G) = Θ(n)

Dudek, Prałat (2015) gave a simple (probabilistic) proof showing that ˆ r (P

) < 137n

Natural question: For the size Ramsey numbers of which graph sequences

do we have linearity in the number of vertices?

Size Ramsey number of P _n

Question (Erd ˝os; 1981) Is it true that lim

n→∞

ˆ r (P

)

n = ∞ and lim

n→∞

ˆ r (P

) n

= 0 ? Theorem (Beck; 1983)

There is a constant C > 0 such that for every n ∈ N it holds that ˆr(P

) < Cn . For short:

ˆ r (P

) = O(n) .

Beck’s proof uses a probabilistic construction (and a large constant C) Alon, Chung (1988) gave an explicit construction of a graph G → P

with e(G) = Θ(n)

Dudek, Prałat (2015) gave a simple (probabilistic) proof showing that ˆ r (P

) < 137n

Natural question: For the size Ramsey numbers of which graph sequences

do we have linearity in the number of vertices?

Linearity of the size Ramsey number

Linearity of the size Ramsey number is known for ... the path P

the cycle C

trees T on n vertices of bounded maximum degree ∆(T ) ≤ k

Question (Beck; 1990)

Is ˆ r (F ) linear for all graphs with bounded maximum degree? Theorem (Rödl, Szemerédi; 2000)

There is a sequence of graphs H with v (H) = n, ∆(H) = 3 and such that ˆ r (H) = Ω(n log

n) .

Theorem (Kohayakawa, Rödl, Schacht, Szemerédi; 2011)

For every ∆ ∈ N there is a constant c = c(∆) such that the following holds: For every graph F with v (F ) = n and maximum degree ∆ it holds that

ˆ r (F ) ≤ cn

log

n .

Linearity of the size Ramsey number

Linearity of the size Ramsey number is known for ...

the path P

the cycle C

trees T on n vertices of bounded maximum degree ∆(T ) ≤ k Question (Beck; 1990)

Is ˆ r (F ) linear for all graphs with bounded maximum degree? Theorem (Rödl, Szemerédi; 2000)

There is a sequence of graphs H with v (H) = n, ∆(H) = 3 and such that ˆ r (H) = Ω(n log

n) .

Theorem (Kohayakawa, Rödl, Schacht, Szemerédi; 2011)

For every ∆ ∈ N there is a constant c = c(∆) such that the following holds: For every graph F with v (F ) = n and maximum degree ∆ it holds that

ˆ r (F ) ≤ cn

log

n .

Linearity of the size Ramsey number

Linearity of the size Ramsey number is known for ...

the path P

the cycle C

trees T on n vertices of bounded maximum degree ∆(T ) ≤ k Question (Beck; 1990)

Is ˆ r (F ) linear for all graphs with bounded maximum degree? Theorem (Rödl, Szemerédi; 2000)

There is a sequence of graphs H with v (H) = n, ∆(H) = 3 and such that ˆ r (H) = Ω(n log

n) .

Theorem (Kohayakawa, Rödl, Schacht, Szemerédi; 2011)

For every ∆ ∈ N there is a constant c = c(∆) such that the following holds: For every graph F with v (F ) = n and maximum degree ∆ it holds that

ˆ r (F ) ≤ cn

log

n .

Linearity of the size Ramsey number

Linearity of the size Ramsey number is known for ...

the path P

the cycle C

proved by Haxell, Kohayakawa, Łuczak (2008) for the induced size Ramsey number

using regularity methods (very large constant) proved by Javadi, Khoeini, Omidi, Pokrovskiy (2017+) without the regularity lemma:

ˆ r (C

) ≤

( 113482 · 10

n if n is odd 2514 · 10

n if n is even for large enough n.

trees T on n vertices of bounded maximum degree ∆(T ) ≤ k Question (Beck; 1990)

Is ˆ r (F ) linear for all graphs with bounded maximum degree? Theorem (Rödl, Szemerédi; 2000)

There is a sequence of graphs H with v (H) = n, ∆(H) = 3 and such that ˆ r (H) = Ω(n log

n) .

Theorem (Kohayakawa, Rödl, Schacht, Szemerédi; 2011)

For every ∆ ∈ N there is a constant c = c(∆) such that the following holds: For every graph F with v (F ) = n and maximum degree ∆ it holds that

ˆ r (F ) ≤ cn

log

n .

Linearity of the size Ramsey number

Linearity of the size Ramsey number is known for ...

the path P

the cycle C

proved by Haxell, Kohayakawa, Łuczak (2008) for the induced size Ramsey number using regularity methods (very large constant)

proved by Javadi, Khoeini, Omidi, Pokrovskiy (2017+) without the regularity lemma:

ˆ r (C

) ≤

( 113482 · 10

n if n is odd 2514 · 10

n if n is even for large enough n.

trees T on n vertices of bounded maximum degree ∆(T ) ≤ k Question (Beck; 1990)

Is ˆ r (F ) linear for all graphs with bounded maximum degree? Theorem (Rödl, Szemerédi; 2000)

There is a sequence of graphs H with v (H) = n, ∆(H) = 3 and such that ˆ r (H) = Ω(n log

n) .

Theorem (Kohayakawa, Rödl, Schacht, Szemerédi; 2011)

For every ∆ ∈ N there is a constant c = c(∆) such that the following holds: For every graph F with v (F ) = n and maximum degree ∆ it holds that

ˆ r (F ) ≤ cn

log

n .

Linearity of the size Ramsey number

Linearity of the size Ramsey number is known for ...

the path P

the cycle C

proved by Haxell, Kohayakawa, Łuczak (2008) for the induced size Ramsey number using regularity methods (very large constant) proved by Javadi, Khoeini, Omidi, Pokrovskiy (2017+) without the regularity lemma

: ˆ r (C

) ≤

( 113482 · 10

n if n is odd 2514 · 10

n if n is even for large enough n.

trees T on n vertices of bounded maximum degree ∆(T ) ≤ k Question (Beck; 1990)

Is ˆ r (F ) linear for all graphs with bounded maximum degree? Theorem (Rödl, Szemerédi; 2000)

There is a sequence of graphs H with v (H) = n, ∆(H) = 3 and such that ˆ r (H) = Ω(n log

n) .

Theorem (Kohayakawa, Rödl, Schacht, Szemerédi; 2011)

For every ∆ ∈ N there is a constant c = c(∆) such that the following holds: For every graph F with v (F ) = n and maximum degree ∆ it holds that

ˆ r (F ) ≤ cn

log

n .

Linearity of the size Ramsey number

Linearity of the size Ramsey number is known for ...

the path P

the cycle C

proved by Haxell, Kohayakawa, Łuczak (2008) for the induced size Ramsey number using regularity methods (very large constant) proved by Javadi, Khoeini, Omidi, Pokrovskiy (2017+) without the regularity lemma:

ˆ r (C

) ≤

( 113482 · 10

n if n is odd 2514 · 10

n if n is even for large enough n.

trees T on n vertices of bounded maximum degree ∆(T ) ≤ k Question (Beck; 1990)

Is ˆ r (F ) linear for all graphs with bounded maximum degree? Theorem (Rödl, Szemerédi; 2000)

There is a sequence of graphs H with v (H) = n, ∆(H) = 3 and such that ˆ r (H) = Ω(n log

n) .

Theorem (Kohayakawa, Rödl, Schacht, Szemerédi; 2011)

For every ∆ ∈ N there is a constant c = c(∆) such that the following holds: For every graph F with v (F ) = n and maximum degree ∆ it holds that

ˆ r (F ) ≤ cn

log

n .

Linearity of the size Ramsey number

Linearity of the size Ramsey number is known for ...

the path P

the cycle C

trees T on n vertices of bounded maximum degree ∆(T ) ≤ k Question (Beck; 1990)

Is ˆ r (F ) linear for all graphs with bounded maximum degree? Theorem (Rödl, Szemerédi; 2000)

There is a sequence of graphs H with v (H) = n, ∆(H) = 3 and such that ˆ r (H) = Ω(n log

n) .

Theorem (Kohayakawa, Rödl, Schacht, Szemerédi; 2011)

For every ∆ ∈ N there is a constant c = c(∆) such that the following holds: For every graph F with v (F ) = n and maximum degree ∆ it holds that

ˆ r (F ) ≤ cn

log

n .

Linearity of the size Ramsey number

Linearity of the size Ramsey number is known for ...

the path P

the cycle C

trees T on n vertices of bounded maximum degree ∆(T ) ≤ k

Question (Beck; 1990)

Is ˆ r (F ) linear for all graphs with bounded maximum degree? Theorem (Rödl, Szemerédi; 2000)

There is a sequence of graphs H with v (H) = n, ∆(H) = 3 and such that ˆ r (H) = Ω(n log

n) .

Theorem (Kohayakawa, Rödl, Schacht, Szemerédi; 2011)

For every ∆ ∈ N there is a constant c = c(∆) such that the following holds: For every graph F with v (F ) = n and maximum degree ∆ it holds that

ˆ r (F ) ≤ cn

log

n .

Linearity of the size Ramsey number

Linearity of the size Ramsey number is known for ...

the path P

the cycle C

trees T on n vertices of bounded maximum degree ∆(T ) ≤ k almost proved by Beck (1983):

ˆ r (T ) = O(kn log

n) proved by Friedman, Pippenger (1987)

both papers: universality result Question (Beck; 1990)

Is ˆ r (F ) linear for all graphs with bounded maximum degree? Theorem (Rödl, Szemerédi; 2000)

There is a sequence of graphs H with v (H) = n, ∆(H) = 3 and such that ˆ r (H) = Ω(n log

n) .

Theorem (Kohayakawa, Rödl, Schacht, Szemerédi; 2011)

For every ∆ ∈ N there is a constant c = c(∆) such that the following holds: For every graph F with v (F ) = n and maximum degree ∆ it holds that

ˆ r (F ) ≤ cn

log

n .

Linearity of the size Ramsey number

Linearity of the size Ramsey number is known for ...

the path P

the cycle C

trees T on n vertices of bounded maximum degree ∆(T ) ≤ k almost proved by Beck (1983):

ˆ r (T ) = O(kn log

n) proved by Friedman, Pippenger (1987) both papers: universality result

Question (Beck; 1990)

Is ˆ r (F ) linear for all graphs with bounded maximum degree? Theorem (Rödl, Szemerédi; 2000)

There is a sequence of graphs H with v (H) = n, ∆(H) = 3 and such that ˆ r (H) = Ω(n log

n) .

Theorem (Kohayakawa, Rödl, Schacht, Szemerédi; 2011)

For every ∆ ∈ N there is a constant c = c(∆) such that the following holds: For every graph F with v (F ) = n and maximum degree ∆ it holds that

ˆ r (F ) ≤ cn

log

n .

Linearity of the size Ramsey number

Linearity of the size Ramsey number is known for ...

the path P

the cycle C

trees T on n vertices of bounded maximum degree ∆(T ) ≤ k

Question (Beck; 1990)

Is ˆ r (F ) linear for all graphs with bounded maximum degree? Theorem (Rödl, Szemerédi; 2000)

There is a sequence of graphs H with v (H) = n, ∆(H) = 3 and such that ˆ r (H) = Ω(n log

n) .

Theorem (Kohayakawa, Rödl, Schacht, Szemerédi; 2011)

For every ∆ ∈ N there is a constant c = c(∆) such that the following holds: For every graph F with v (F ) = n and maximum degree ∆ it holds that

ˆ r (F ) ≤ cn

log

n .

Linearity of the size Ramsey number

Linearity of the size Ramsey number is known for ...

the path P

the cycle C

trees T on n vertices of bounded maximum degree ∆(T ) ≤ k Question (Beck; 1990)

Is ˆ r (F ) linear for all graphs with bounded maximum degree?

Theorem (Rödl, Szemerédi; 2000)

There is a sequence of graphs H with v (H) = n, ∆(H) = 3 and such that ˆ r (H) = Ω(n log

n) .

Theorem (Kohayakawa, Rödl, Schacht, Szemerédi; 2011)

For every ∆ ∈ N there is a constant c = c(∆) such that the following holds: For every graph F with v (F ) = n and maximum degree ∆ it holds that

ˆ r (F ) ≤ cn

log

n .

Linearity of the size Ramsey number

Linearity of the size Ramsey number is known for ...

the path P

the cycle C

trees T on n vertices of bounded maximum degree ∆(T ) ≤ k Question (Beck; 1990)

Is ˆ r (F ) linear for all graphs with bounded maximum degree?

Theorem (Rödl, Szemerédi; 2000)

There is a sequence of graphs H with v (H) = n, ∆(H) = 3 and such that ˆ r (H) = Ω(n log

n) .

Theorem (Kohayakawa, Rödl, Schacht, Szemerédi; 2011)

For every ∆ ∈ N there is a constant c = c(∆) such that the following holds: For every graph F with v (F ) = n and maximum degree ∆ it holds that

ˆ r (F ) ≤ cn

log

n .

Linearity of the size Ramsey number

Linearity of the size Ramsey number is known for ...

the path P

the cycle C

trees T on n vertices of bounded maximum degree ∆(T ) ≤ k Question (Beck; 1990)

Is ˆ r (F ) linear for all graphs with bounded maximum degree?

Theorem (Rödl, Szemerédi; 2000)

There is a sequence of graphs H with v (H) = n, ∆(H) = 3 and such that ˆ r (H) = Ω(n log

n) .

Theorem (Kohayakawa, Rödl, Schacht, Szemerédi; 2011)

For every ∆ ∈ N there is a constant c = c(∆) such that the following holds:

For every graph F with v (F ) = n and maximum degree ∆ it holds that

ˆ r (F ) ≤ cn

log

n .

Powers of paths

Definition: Let F = (V , E) be a graph. The k -th power F

of F is defined on the same vertex set as F with edges given as follows:

uv ∈ E(F

) : ⇔ dist

(u, v ) ≤ k

P

Problem (Conlon; 2016)

Find out whether the size Ramsey number of P

is linear in n.

Theorem (C., Jenssen, Kohayakawa, Morrison, Mota, Reding, Roberts; 2017+)

ˆ r (P

) = O(n)

and ˆ r (C

) = O(n)

Powers of paths

Definition: Let F = (V , E) be a graph. The k -th power F

of F is defined on the same vertex set as F with edges given as follows:

uv ∈ E(F

) : ⇔ dist

(u, v ) ≤ k

P

Problem (Conlon; 2016)

Find out whether the size Ramsey number of P

is linear in n.

Theorem (C., Jenssen, Kohayakawa, Morrison, Mota, Reding, Roberts; 2017+)

ˆ r (P

) = O(n)

and ˆ r (C

) = O(n)

Powers of paths

Definition: Let F = (V , E) be a graph. The k -th power F

of F is defined on the same vertex set as F with edges given as follows:

uv ∈ E(F

) : ⇔ dist

(u, v ) ≤ k

P

Problem (Conlon; 2016)

Find out whether the size Ramsey number of P

is linear in n.

Theorem (C., Jenssen, Kohayakawa, Morrison, Mota, Reding, Roberts; 2017+)

ˆ r (P

) = O(n)

and ˆ r (C

) = O(n)

Powers of paths

Definition: Let F = (V , E) be a graph. The k -th power F

of F is defined on the same vertex set as F with edges given as follows:

uv ∈ E(F

) : ⇔ dist

(u, v ) ≤ k

P

Problem (Conlon; 2016)

Find out whether the size Ramsey number of P

is linear in n.

Theorem (C., Jenssen, Kohayakawa, Morrison, Mota, Reding, Roberts; 2017+)

ˆ r (P

) = O(n)

and ˆ r (C

) = O(n)

Powers of paths

Definition: Let F = (V , E) be a graph. The k -th power F

of F is defined on the same vertex set as F with edges given as follows:

uv ∈ E(F

) : ⇔ dist

(u, v ) ≤ k

P

Problem (Conlon; 2016)

Find out whether the size Ramsey number of P

is linear in n.

Theorem (C., Jenssen, Kohayakawa, Morrison, Mota, Reding, Roberts; 2017+)

ˆ r (P

) = O(n)

and ˆ r (C

) = O(n)

Powers of paths

Definition: Let F = (V , E) be a graph. The k -th power F

of F is defined on the same vertex set as F with edges given as follows:

uv ∈ E(F

) : ⇔ dist

(u, v ) ≤ k

P

Problem (Conlon; 2016)

Find out whether the size Ramsey number of P

is linear in n.

Theorem (C., Jenssen, Kohayakawa, Morrison, Mota, Reding, Roberts; 2017+)

ˆ r (P

) = O(n)

and ˆ r (C

) = O(n)

Powers of paths

Definition: Let F = (V , E) be a graph. The k -th power F

of F is defined on the same vertex set as F with edges given as follows:

uv ∈ E(F

) : ⇔ dist

(u, v ) ≤ k

P

Problem (Conlon; 2016)

Find out whether the size Ramsey number of P

is linear in n.

Theorem (C., Jenssen, Kohayakawa, Morrison, Mota, Reding, Roberts;

2017+)

ˆ r (P

) = O(n)

and ˆ r (C

) = O(n)

Powers of paths

Definition: Let F = (V , E) be a graph. The k -th power F

of F is defined on the same vertex set as F with edges given as follows:

uv ∈ E(F

) : ⇔ dist

(u, v ) ≤ k

P

Problem (Conlon; 2016)

Find out whether the size Ramsey number of P

is linear in n.

Theorem (C., Jenssen, Kohayakawa, Morrison, Mota, Reding, Roberts;

2017+)

ˆ r (P

) = O(n) and ˆ r (C

) = O(n)

Proof for ˆ r (P _n ^k ) = O(n)

Proof for ˆ r (P _n ^k ) = O(n)

with ∆(H) = Θ(1) and the property

”for every disjoint sets S, T ⊂ V (H) with

|S|, |T | ≥ γn it holds that e

(S, T ) > 0”

start with a graph H on Θ(n) vertices,

Proof for ˆ r (P _n ^k ) = O(n)

H

with ∆(H) = Θ(1) and the property

”for every disjoint sets S, T ⊂ V (H) with

|S|, |T | ≥ γn it holds that e

(S, T ) > 0”

start with a graph H on Θ(n) vertices,

(between large sets there is always an edge)

Proof for ˆ r (P _n ^k ) = O(n)

H start with a graph H on Θ(n) vertices, with ∆(H) = Θ(1) and the property

”for every disjoint sets S, T ⊂ V (H) with

|S|, |T | ≥ γn it holds that e

(S, T ) > 0”

take k-the power H

of H

is always an edge)

Proof for ˆ r (P _n ^k ) = O(n)

H start with a graph H on Θ(n) vertices, with ∆(H) = Θ(1) and the property

”for every disjoint sets S, T ⊂ V (H) with

|S|, |T | ≥ γn it holds that e

(S, T ) > 0”

H

take k-the power H

of H

is always an edge)

Proof for ˆ r (P _n ^k ) = O(n)

H start with a graph H on Θ(n) vertices, with ∆(H) = Θ(1) and the property

”for every disjoint sets S, T ⊂ V (H) with

|S|, |T | ≥ γn it holds that e

(S, T ) > 0”

H

take k-the power H

of H

(it will turn out later that H

→ (P

, P

))

is always an edge)

Proof for ˆ r (P _n ^k ) = O(n)

H

H

start with a graph H on Θ(n) vertices, with ∆(H) = Θ(1) and the property

”for every disjoint sets S, T ⊂ V (H) with

|S|, |T | ≥ γn it holds that e

(S, T ) > 0”

take k-the power H

of H

(it will turn out later that H

→ (P

, P

))

take ”complete-t-blow-up” (H

)

of H

:

is always an edge)

Proof for ˆ r (P _n ^k ) = O(n)

H

H

start with a graph H on Θ(n) vertices, with ∆(H) = Θ(1) and the property

”for every disjoint sets S, T ⊂ V (H) with

|S|, |T | ≥ γn it holds that e

(S, T ) > 0”

take k-the power H

of H

(it will turn out later that H

→ (P

, P

))

take ”complete-t-blow-up” (H

)

of H

:

is always an edge)

Proof for ˆ r (P _n ^k ) = O(n)

H

H

start with a graph H on Θ(n) vertices, with ∆(H) = Θ(1) and the property

”for every disjoint sets S, T ⊂ V (H) with

|S|, |T | ≥ γn it holds that e

(S, T ) > 0”

take k-the power H

of H

(it will turn out later that H

→ (P

, P

))

take ”complete-t-blow-up” (H

)

of H

:

• replace each vertex by a clique of size r(K

), called cluster

is always an edge)

Proof for ˆ r (P _n ^k ) = O(n)

H

H

start with a graph H on Θ(n) vertices, with ∆(H) = Θ(1) and the property

”for every disjoint sets S, T ⊂ V (H) with

|S|, |T | ≥ γn it holds that e

(S, T ) > 0”

take k-the power H

of H

(it will turn out later that H

→ (P

, P

))

take ”complete-t-blow-up” (H

)

of H

:

• replace each vertex by a clique

• replace each edge by a complete bipartite graph

(H

)

is always an edge)

of size r(K

), called cluster

Proof for ˆ r (P _n ^k ) = O(n)

H

H

(H

)

start with a graph H on Θ(n) vertices, with ∆(H) = Θ(1) and the property

”for every disjoint sets S, T ⊂ V (H) with

|S|, |T | ≥ γn it holds that e

(S, T ) > 0”

take k-the power H

of H

(it will turn out later that H

→ (P

, P

))

take ”complete-t-blow-up” (H

)

of H

:

• replace each vertex by a clique

• replace each edge by a complete bipartite graph

Note: e((H

)

) = O(n)

(t = t(k) is chosen as a large constant) (between large sets there

is always an edge)

of size r(K

), called cluster

Proof for ˆ r (P _n ^k ) = O(n)

H

H

(H

)

start with a graph H on Θ(n) vertices, with ∆(H) = Θ(1) and the property

”for every disjoint sets S, T ⊂ V (H) with

|S|, |T | ≥ γn it holds that e

(S, T ) > 0”

take k-the power H

of H

(it will turn out later that H

→ (P

, P

))

take ”complete-t-blow-up” (H

)

of H

:

• replace each vertex by a clique

• replace each edge by a complete bipartite graph

Theorem:

(H

)

is Ramsey for P

.

is always an edge)

of size r(K

), called cluster

Proof for ˆ r (P _n ^k ) = O(n)

H

H

(H

)

Theorem:

(H

)

is Ramsey for P

.

take ”complete-t-blow-up” (H

)

of H

:

• replace each vertex by a clique

• replace each edge by a complete bipartite graph

(between large sets there is always an edge)

of size r(K

), called cluster

Proof for ˆ r (P _n ^k ) = O(n)

H

H

(H

)

Theorem:

(H

)

is Ramsey for P

.

take ”complete-t-blow-up” (H

)

of H

:

• replace each vertex by a clique

• replace each edge by a complete bipartite graph

colouring χ

(between large sets there is always an edge)

Proof: Let a 2-colouring χ of E((H

)

) be a given.

of size r(K

), called cluster

Proof for ˆ r (P _n ^k ) = O(n)

H

H

(H

)

Theorem:

(H

)

is Ramsey for P

.

In each cluster find a m.c. copy of K

.

take ”complete-t-blow-up” (H

)

of H

:

• replace each vertex by a clique

• replace each edge by a complete bipartite graph

colouring χ

(between large sets there is always an edge)

Proof: Let a 2-colouring χ of E((H

)

) be a given.

of size r(K

), called cluster

Proof for ˆ r (P _n ^k ) = O(n)

Theorem:

(H

)

is Ramsey for P

.

Proof: Let a 2-colouring χ of E((H

)

) be a given.

In each cluster find a m.c. copy of K

.

take ”complete-t-blow-up” (H

)

of H

:

• replace each vertex by a clique

• replace each edge by a complete bipartite graph

H

H

(H

)

colouring χ

(between large sets there is always an edge)

of size r(K

), called cluster

Proof for ˆ r (P _n ^k ) = O(n)

Theorem:

(H

)

is Ramsey for P

.

In each cluster find a m.c. copy of K

.

H

H

(H

)

colouring χ

(between large sets there is always an edge)

Proof: Let a 2-colouring χ of E((H

)

) be a given.

Proof for ˆ r (P _n ^k ) = O(n)

Theorem:

(H

)

is Ramsey for P

.

In each cluster find a m.c. copy of K

.

H

H

(H

)

W.l.o.g. half of all clusters have a blue copy of K

.

colouring χ

(between large sets there is always an edge)

Proof: Let a 2-colouring χ of E((H

)

) be a given.

Proof for ˆ r (P _n ^k ) = O(n)

Theorem:

(H

)

is Ramsey for P

.

In each cluster find a m.c. copy of K

.

H

H

(H

)

W.l.o.g. half of all clusters have a blue copy of K

. Set W := {v ∈ V (H) : cluster C(v) has blue K

}, reduce: H to F := H[W ]

H

to F

(H

)

to F

induced by the blues K

s

colouring χ

(between large sets there is always an edge)

Proof: Let a 2-colouring χ of E((H

)

) be a given.

Proof for ˆ r (P _n ^k ) = O(n)

H

H

Theorem:

(H

)

is Ramsey for P

.

In each cluster find a m.c. copy of K

.

W.l.o.g. half of all clusters have a blue copy of K

. Set W := {v ∈ V (H) : cluster C(v) has blue K

}, reduce: H to F := H[W ]

H

to F

F

⊂ (H

)

colouring χ

(H

)

to F

induced by the blues K

s

is always an edge)

Proof: Let a 2-colouring χ of E((H

)

) be a given.

Proof for ˆ r (P _n ^k ) = O(n)

F ⊂ H

F

⊂ H

F

⊂ (H

)

Theorem:

(H

)

is Ramsey for P

.

In each cluster find a m.c. copy of K

.

W.l.o.g. half of all clusters have a blue copy of K

. Set W := {v ∈ V (H) : cluster C(v) has blue K

}, reduce: H to F := H[W ]

H

to F

colouring χ

(H

)

to F

induced by the blues K

s

is always an edge)

Proof: Let a 2-colouring χ of E((H

)

) be a given.

Proof for ˆ r (P _n ^k ) = O(n)

F ⊂ H

F

⊂ H

F

⊂ (H

)

Theorem:

(H

)

is Ramsey for P

.

In each cluster find a m.c. copy of K

.

W.l.o.g. half of all clusters have a blue copy of K

. Set W := {v ∈ V (H) : cluster C(v) has blue K

}, reduce: H to F := H[W ]

H

to F

colouring χ

Define an auxiliary colouring χ

of E(F

):

colouring χ

(H

)

to F

induced by the blues K

s

is always an edge)

Proof: Let a 2-colouring χ of E((H

)

) be a given.

Proof for ˆ r (P _n ^k ) = O(n)

F ⊂ H

F

⊂ H

F

⊂ (H

)

Theorem:

(H

)

is Ramsey for P

.

In each cluster find a m.c. copy of K

.

W.l.o.g. half of all clusters have a blue copy of K

. Set W := {v ∈ V (H) : cluster C(v) has blue K

}, reduce: H to F := H[W ]

H

to F

colouring χ

Define an auxiliary colouring χ

of E(F

):

colour e = uv blue if between the corresponding blue K

-copies in F

there is a blue copy of K

, otherwise colour e = uv red.

colouring χ

(H

)

to F

induced by the blues K

s

is always an edge)

Proof: Let a 2-colouring χ of E((H

)

) be a given.

Proof for ˆ r (P _n ^k ) = O(n)

F

⊂ (H

)

F ⊂ H

F

⊂ H

colouring χ

Theorem:

(H

)

is Ramsey for P

.

In each cluster find a m.c. copy of K

.

W.l.o.g. half of all clusters have a blue copy of K

. Set W := {v ∈ V (H) : cluster C(v) has blue K

}, reduce: H to F := H[W ]

H

to F

Define an auxiliary colouring χ

of E(F

):

colouring χ

colour e = uv blue if between the corresponding blue K

-copies in F

there is a blue copy of K

, otherwise colour e = uv red.

(H

)

to F

induced by the blues K

s

is always an edge)

Proof: Let a 2-colouring χ of E((H

)

) be a given.

Proof for ˆ r (P _n ^k ) = O(n)

F

⊂ (H

)

F ⊂ H

F

⊂ H

colouring χ

Theorem:

(H

)

is Ramsey for P

.

In each cluster find a m.c. copy of K

.

W.l.o.g. half of all clusters have a blue copy of K

. Set W := {v ∈ V (H) : cluster C(v) has blue K

}, reduce: H to F := H[W ]

H

to F

Define an auxiliary colouring χ

of E(F

):

colouring χ

colour e = uv blue if between the corresponding blue K

-copies in F

there is a blue copy of K

, otherwise colour e = uv red.

(H

)

to F

induced by the blues K

s

is always an edge)

Proof: Let a 2-colouring χ of E((H

)

) be a given.

Proof for ˆ r (P _n ^k ) = O(n)

F

⊂ (H

)

F ⊂ H

F

⊂ H

colouring χ

Theorem:

(H

)

is Ramsey for P

.

In each cluster find a m.c. copy of K

.

W.l.o.g. half of all clusters have a blue copy of K

. Set W := {v ∈ V (H) : cluster C(v) has blue K

}, reduce: H to F := H[W ]

H

to F

Define an auxiliary colouring χ

of E(F

):

colouring χ

colour e = uv blue if between the corresponding blue K

-copies in F

there is a blue copy of K

, otherwise colour e = uv red.

(H

)

to F

induced by the blues K

s

is always an edge)

Proof: Let a 2-colouring χ of E((H

)

) be a given.

Proof for ˆ r (P _n ^k ) = O(n)

no blue K2k,2k

F

⊂ (H

)

F ⊂ H

F

⊂ H

colouring χ

Theorem:

(H

)

is Ramsey for P

.

In each cluster find a m.c. copy of K

.

W.l.o.g. half of all clusters have a blue copy of K

. Set W := {v ∈ V (H) : cluster C(v) has blue K

}, reduce: H to F := H[W ]

H

to F

Define an auxiliary colouring χ

of E(F

):

colouring χ

colour e = uv blue if between the corresponding blue K

-copies in F

there is a blue copy of K

, otherwise colour e = uv red.

(H

)

to F

induced by the blues K

s

is always an edge)

Proof: Let a 2-colouring χ of E((H

)

) be a given.

Proof for ˆ r (P _n ^k ) = O(n)

F

⊂ (H

)

F ⊂ H

F

⊂ H

colouring χ

Theorem:

(H

)

is Ramsey for P

.

In each cluster find a m.c. copy of K

.

W.l.o.g. half of all clusters have a blue copy of K

. Set W := {v ∈ V (H) : cluster C(v) has blue K

}, reduce: H to F := H[W ]

H

to F

Define an auxiliary colouring χ

of E(F

):

colouring χ

colour e = uv blue if between the corresponding blue K

-copies in F

there is a blue copy of K

, otherwise colour e = uv red.

no blue K2k,2k

(H

)

to F

induced by the blues K

s

is always an edge)

Proof: Let a 2-colouring χ of E((H

)

) be a given.

Proof for ˆ r (P _n ^k ) = O(n)

F

⊂ (H

)

F ⊂ H

F

⊂ H

colouring χ colouring χ

no blue K2k,2k

(between large sets there is always an edge)

Proof for ˆ r (P _n ^k ) = O(n)

Any 2-colouring χ

of E(F

) has a blue copy of P

or a red copy of P

.

F

⊂ (H

)

F ⊂ H

F

⊂ H

colouring χ colouring χ

no blue K2k,2k

Lemma 1 (Key Lemma):

(between large sets there is always an edge)

Proof for ˆ r (P _n ^k ) = O(n)

Any 2-colouring χ

of E(F

) has a blue copy of P

or a red copy of P

.

Lemma 2:

If χ

on E(F

) has a blue copy of P

, then χ has a blue copy of P

in H

. Lemma 3:

If χ

on E(F

) has a red copy of P

, then χ has a red copy of P

in H

. F

⊂ (H

)

F ⊂ H

F

⊂ H

colouring χ colouring χ

no blue K2k,2k

Lemma 1 (Key Lemma):

(between large sets there is always an edge)

Proof for ˆ r (P _n ^k ) = O(n)

Any 2-colouring χ

of E(F

) has a blue copy of P

or a red copy of P

.

Lemma 2:

If χ

on E(F

) has a blue copy of P

, then χ has a blue copy of P

in H

. Lemma 3:

If χ

on E(F

) has a red copy of P

, then χ has a red copy of P

in H

. F

⊂ (H

)

F ⊂ H

F

⊂ H

colouring χ colouring χ

no blue K2k,2k

simple

Lemma 1 (Key Lemma):

(between large sets there is always an edge)

embed- ding

Proof for ˆ r (P _n ^k ) = O(n)

Any 2-colouring χ

of E(F

) has a blue copy of P

or a red copy of P

.

Lemma 2:

If χ

on E(F

) has a blue copy of P

, then χ has a blue copy of P

in H

. Lemma 3:

If χ

on E(F

) has a red copy of P

, then χ has a red copy of P

in H

. F

⊂ (H

)

F ⊂ H

F

⊂ H

colouring χ colouring χ

no blue K2k,2k

simple

greedy embedding

or Lov´asz LL

Lemma 1 (Key Lemma):

(between large sets there is always an edge)

embed- ding

Proof for ˆ r (P _n ^k ) = O(n)

F

⊂ (H

)

F ⊂ H

F

⊂ H

colouring χ colouring χ

no blue K2k,2k

(between large sets there is always an edge)

Any 2-colouring χ

of E(F

) has a

blue copy of P

or a red copy of P

.

Lemma 1 (Key Lemma):

Proof for ˆ r (P _n ^k ) = O(n)

Any 2-colouring χ

of E(F

) has a blue copy of P

or a red copy of P

.

F

⊂ (H

)

F ⊂ H

F

⊂ H

colouring χ colouring χ

no blue K2k,2k

Lemma 1 (Key Lemma):

Theorem (Pokrovskiy, 2017):

Let E(K

) be coloured with red/blue. Then V (K

) can be covered by k blue paths and a red balanced complete (k + 1)-partite graph (vertex-disjoint).

(between large sets there is always an edge)

Proof for ˆ r (P _n ^k ) = O(n)

Any 2-colouring χ

of E(F

) has a blue copy of P

or a red copy of P

.

F

⊂ (H

)

F ⊂ H

F

⊂ H

colouring χ colouring χ

no blue K2k,2k

Lemma 1 (Key Lemma):

Theorem (Pokrovskiy, 2017):

can be covered by k blue paths and a red balanced complete (k + 1)-partite graph (vertex-disjoint).

Proof: Extend χ

by colouring non-edges on V (F

) red.

(between large sets there is always an edge)

Let E(K

) be coloured with red/blue. Then V (K

)

Proof for ˆ r (P _n ^k ) = O(n)

Any 2-colouring χ

of E(F

) has a blue copy of P

or a red copy of P

.

F

⊂ (H

)

F ⊂ H

F

⊂ H

colouring χ colouring χ

no blue K2k,2k

Lemma 1 (Key Lemma):

Theorem (Pokrovskiy, 2017):

can be covered by k blue paths and a red balanced complete (k + 1)-partite graph (vertex-disjoint).

Proof:

Suppose that there is no blue copy of P

.

Extend χ

by colouring non-edges on V (F

) red.

(between large sets there is always an edge)

Let E(K

) be coloured with red/blue. Then V (K

)