on the size ramsey number -...

On the size Ramsey number

Dennis Clemens

Monash University MelbourneMarch 19th 21st 2018

joint work with: (1) Matthew Jenssen, Yoshiharu Kohayakawa, NatashaMorrison, 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 colorsthe edges of G with 2 colors, one of the color classes contains a copy of F .

Ramsey graphs


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

Ramsey graphs



Typical example: K6 → K3 and K5 6→ K3

Ramsey graphs




Theorem (Ramsey; 1930)

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

Ramsey graphs






Typical questions: Given a graph F . Determine

r(F ) := minv(G) : G→ F [Ramsey number]

s(F ) := minδ(G) : G→ F and H 6→ F for all H ( Gr(F ) := mine(G) : G→ F [size Ramsey number]

(... and other graph parameters)

Ramsey graphs








s(F ) := minδ(G) : G→ F and H 6→ F for all H ( G

r(F ) := mine(G) : G→ F [size Ramsey number]


Ramsey graphs








s(F ) := minδ(G) : G→ F and H 6→ F for all H ( Gr(F ) := mine(G) : G→ F [size Ramsey number]


Size Ramsey number

Definition (Erdos, Faudree, Rousseau, Schelp; 1978)

The size Ramsey number of a graph F is defined as

r(F ) := mine(G) : G→ F .

Observation: For every graph F we have

r(F ) ≤(

r(F )

2

).

Erdos et. al. (1978) asked for graphs for which equality holds and for graphsequences for which r(F ) is of smaller order than

(r(F )2

).

Theorem (Chvátal; 1978)

r(Kn) =

(r(Kn)

2

)

Question (Erdos; 1981)

Is it true that limn→∞

r(Pn)

n=∞ and lim

n→∞

r(Pn)

n2 = 0 ?

Size Ramsey number



r(F ) := mine(G) : G→ F .


r(F ) ≤(

r(F )

2

).

Erdos et. al. (1978) asked for graphs for which equality holds

and for graphsequences for which r(F ) is of smaller order than

(r(F )2

).


r(Kn) =

(r(Kn)

2

)



r(Pn)

n=∞ and lim

n→∞

r(Pn)

n2 = 0 ?

Size Ramsey number



r(F ) := mine(G) : G→ F .


r(F ) ≤(

r(F )

2

).

Erdos et. al. (1978) asked for graphs for which equality holds and for graphsequences for which r(F ) is of smaller order than

(r(F )2

).


r(Kn) =

(r(Kn)

2

)



r(Pn)

n=∞ and lim

n→∞

r(Pn)

n2 = 0 ?

Size Ramsey number of Pn



r(Pn)

n=∞ and lim

n→∞

r(Pn)

n2 = 0 ?

Theorem (Beck; 1983)

There is a constant C > 0 such that for every n ∈ N it holds that r(Pn) < Cn .For short:

r(Pn) = 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→ Pn withe(G) = Θ(n)

Dudek, Prałat (2015) gave a simple (probabilistic) proof showing thatr(Pn) < 137n

Natural question: For the size Ramsey numbers of which graph sequencesdo 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 Pn

the cycle Cn

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

Question (Beck; 1990)

Is 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

r(H) = Ω(n log160 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

r(F ) ≤ cn2− 1∆ log

1∆ n .

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

the path Pn

the cycle Cn






r(H) = Ω(n log160 n) .



r(F ) ≤ cn2− 1∆ log

1∆ n .


the path Pn

the cycle Cn

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

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

r(Cn) ≤

113482 · 106n if n is odd2514 · 106n if n is even

for large enough n.






r(H) = Ω(n log160 n) .



r(F ) ≤ cn2− 1∆ log

1∆ n .


the path Pn

the cycle Cn

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

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

r(Cn) ≤


for large enough n.






r(H) = Ω(n log160 n) .



r(F ) ≤ cn2− 1∆ log

1∆ n .


the path Pn

the cycle Cn

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

:

r(Cn) ≤


for large enough n.






r(H) = Ω(n log160 n) .



r(F ) ≤ cn2− 1∆ log

1∆ n .


the path Pn

the cycle Cn

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

r(Cn) ≤


for large enough n.






r(H) = Ω(n log160 n) .



r(F ) ≤ cn2− 1∆ log

1∆ n .


the path Pn

the cycle Cn






r(H) = Ω(n log160 n) .



r(F ) ≤ cn2− 1∆ log

1∆ n .


the path Pn

the cycle Cn

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

r(T ) = O(kn log12 n)

proved by Friedman, Pippenger (1987)

both papers: universality result





r(H) = Ω(n log160 n) .



r(F ) ≤ cn2− 1∆ log

1∆ n .


the path Pn

the cycle Cn






r(H) = Ω(n log160 n) .



r(F ) ≤ cn2− 1∆ log

1∆ n .

Powers of paths

Definition: Let F = (V ,E) be a graph. The k -th power F k of F is defined onthe same vertex set as F with edges given as follows:

uv ∈ E(F k ) :⇔ distF (u, v) ≤ k

P7

Problem (Conlon; 2016)

Find out whether the size Ramsey number of Pkn is linear in n.

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

r(Pkn ) = O(n)

and r(Ckn ) = O(n)

Powers of paths



P 27




r(Pkn ) = O(n)

and r(Ckn ) = O(n)

Powers of paths



P 37




r(Pkn ) = O(n)

and r(Ckn ) = O(n)

Powers of paths



P 37




r(Pkn ) = O(n) and r(Ck

n ) = O(n)

Proof for r(Pkn ) = O(n)


with ∆(H) = Θ(1) and the property

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

|S|, |T | ≥ γn it holds that eH(S, T ) > 0”

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


Hwith ∆(H) = Θ(1) and the property




(between large sets there

is always an edge)


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




take k-the power Hk of H


is always an edge)






Hk(k = 2)



is always an edge)






Hk(k = 2)


(it will turn out later that Hk → (Pn, Pkn ))


is always an edge)


H

Hk(k = 2)







take ”complete-t-blow-up” (Hk)t of Hk:


is always an edge)


H

Hk(k = 2)








• replace each vertex by a clique

of size r(Kt), called cluster


is always an edge)


H

Hk(k = 2)









• replace each edge by a complete

bipartite graph

(Hk)t


is always an edge)



H

Hk(k = 2)

(Hk)t










bipartite graph

Note: e((Hk)t) = O(n)(t = t(k) is chosen as a large constant)


is always an edge)



H

Hk(k = 2)

(Hk)t










bipartite graph

Theorem:

(Hk)t is Ramsey for P kn .


is always an edge)



H

Hk(k = 2)

(Hk)t

Theorem:





bipartite graph


is always an edge)



H

Hk(k = 2)

(Hk)t

Theorem:





bipartite graph

colouring χ


is always an edge)

Proof: Let a 2-colouring χ of E((Hk)t) be a given.



H

Hk(k = 2)

(Hk)t

Theorem:


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




bipartite graph

colouring χ


is always an edge)




Theorem:







bipartite graph

H

Hk(k = 2)

(Hk)t

colouring χ


is always an edge)



Theorem:


In each cluster find a m.c. copy of Kt.H

Hk(k = 2)

(Hk)t

colouring χ


is always an edge)



Theorem:



Hk(k = 2)

(Hk)t

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

colouring χ


is always an edge)



Theorem:



Hk(k = 2)

(Hk)t


Set W := v ∈ V (H) : cluster C(v) has blue Kt,reduce: H to F := H[W ]

Hk to F k

(Hk)t to F ′ induced by the blues K ′ts

colouring χ


is always an edge)



H

Hk(k = 2)

Theorem:





Hk to F k

F ′ ⊂ (Hk)t

colouring χ



is always an edge)



F ⊂ H

F k ⊂ Hk

F ′ ⊂ (Hk)t

Theorem:





Hk to F k

colouring χ



is always an edge)



F ⊂ H

F k ⊂ Hk

F ′ ⊂ (Hk)t

Theorem:





Hk to F k

colouring χ

Define an auxiliary colouring χ′ of E(F k):

colouring χ′(Hk)t to F ′ induced by the blues K ′ts


is always an edge)



F ⊂ H

F k ⊂ Hk

F ′ ⊂ (Hk)t

Theorem:





Hk to F k

colouring χ


colour e = uv blue if between the

corresponding blue Kt-copies in F ′

there is a blue copy of K2k,2k,

otherwise colour e = uv red.

colouring χ′(Hk)t to F ′ induced by the blues K ′ts


is always an edge)



F ′ ⊂ (Hk)t

F ⊂ H

F k ⊂ Hk

colouring χ

Theorem:





Hk to F k


colouring χ′







is always an edge)



no blue K2k,2k

F ′ ⊂ (Hk)t

F ⊂ H

F k ⊂ Hk

colouring χ

Theorem:





Hk to F k


colouring χ′







is always an edge)



F ′ ⊂ (Hk)t

F ⊂ H

F k ⊂ Hk

colouring χ

Theorem:





Hk to F k


colouring χ′





no blue K2k,2k



is always an edge)



F ′ ⊂ (Hk)t

F ⊂ H

F k ⊂ Hk

colouring χ

colouring χ′

no blue K2k,2k


is always an edge)


Any 2-colouring χ′ of E(F k) has a

blue copy of Pn or a red copy of P kn .

F ′ ⊂ (Hk)t

F ⊂ H

F k ⊂ Hk

colouring χ

colouring χ′

no blue K2k,2k

Lemma 1 (Key Lemma):


is always an edge)




Lemma 2:

If χ′ on E(F k) has a blue copy of Pn,

then χ has a blue copy of P kn in Hk

t .

Lemma 3:

If χ′ on E(F k) has a red copy of P kn ,

then χ has a red copy of P kn in Hk

t .

F ′ ⊂ (Hk)t

F ⊂ H

F k ⊂ Hk

colouring χ

colouring χ′

no blue K2k,2k



is always an edge)




Lemma 2:



t .

Lemma 3:



t .

F ′ ⊂ (Hk)t

F ⊂ H

F k ⊂ Hk

colouring χ

colouring χ′

no blue K2k,2k

simple



is always an edge)

embed-ding




Lemma 2:



t .

Lemma 3:



t .

F ′ ⊂ (Hk)t

F ⊂ H

F k ⊂ Hk

colouring χ

colouring χ′

no blue K2k,2k

simple

greedyembedding

orLovasz LL



is always an edge)

embed-ding


F ′ ⊂ (Hk)t

F ⊂ H

F k ⊂ Hk

colouring χ

colouring χ′

no blue K2k,2k


is always an edge)







F ′ ⊂ (Hk)t

F ⊂ H

F k ⊂ Hk

colouring χ

colouring χ′

no blue K2k,2k


Theorem (Pokrovskiy, 2017):

Let E(Km) be coloured with red/blue. Then V (Km)

can be covered by k blue paths and a red balanced

complete (k + 1)-partite graph (vertex-disjoint).


is always an edge)




F ′ ⊂ (Hk)t

F ⊂ H

F k ⊂ Hk

colouring χ

colouring χ′

no blue K2k,2k





Proof: Extend χ′ by colouring non-edges on V (F k) red.


is always an edge)





F ′ ⊂ (Hk)t

F ⊂ H

F k ⊂ Hk

colouring χ

colouring χ′

no blue K2k,2k





Proof:

Suppose that there is no blue copy of Pn.

Extend χ′ by colouring non-edges on V (F k) red.


is always an edge)





F ′ ⊂ (Hk)t

F ⊂ H

F k ⊂ Hk

colouring χ

colouring χ′

no blue K2k,2k





Proof:



By Pokrovskiys Theorem find a huge red balanced

complete (k + 1)-partite graph.


is always an edge)





F ′ ⊂ (Hk)t

F ⊂ H

F k ⊂ Hk

colouring χ

colouring χ′

no blue K2k,2k





Proof:




complete (k + 1)-partite graph. Using only the edges

find a copy of Pn.from F ⊂ H


is always an edge)





F ′ ⊂ (Hk)t

F ⊂ H

F k ⊂ Hk

colouring χ

colouring χ′

no blue K2k,2k





Proof:




complete (k + 1)-partite graph. Using only the edges

find a copy of Pn.from F ⊂ H


is always an edge)

Then use the edges from F k to

finish a red copy of P kn .


Open problems and work in progress

Problem 1 (What about grids?): Let Gk,n be the grid graph of size k × n, i.e.the cartesian product of the paths Pk and Pn. If k is a fixed constant, then theprevious result implies

r(Gk,n) = O(n) .

Now, let Gdn be the d-dimensional grid, obtained by taking the cartesian

product of d copies of Pn.

Question (CJKMMRR; 2017+)

For any d ≥ 2, is it true that r(Gdn ) = O(nd )?

Theorem (C., Miralaei, Reding, Schacht, Taraz; 2018+)

r(G2n) = O(n3+o(1))


Problem 1 (What about grids?): Let Gk,n be the grid graph of size k × n, i.e.the cartesian product of the paths Pk and Pn.

If k is a fixed constant, then theprevious result implies

r(Gk,n) = O(n) .






r(G2n) = O(n3+o(1))



r(Gk,n) = O(n) .






r(G2n) = O(n3+o(1))



r(Gk,n) = O(n) .



G25 G3

3




r(G2n) = O(n3+o(1))



r(Gk,n) = O(n) .






r(G2n) = O(n3+o(1))


Problem 1 (What about grids?): Let Gdn be the d-dimensional grid, obtained

by taking the cartesian product of d copies of Pn.




r(G2n) = O(n3+o(1))


Problem 1 (What about grids?): Let Gdn be the d-dimensional grid, obtained

by taking the cartesian product of d copies of Pn.




r(G2n) = O(n3+o(1))

Problem 2 (What about more than 2 colours?):Denote with rq(F ) the size Ramsey number in case of colourings with qcolours. It is still true that

rq(Pkn ) = O(n) ?

Problem 3 (What about hypergraphs?):How large is the size Ramsey number of tight paths and tight cycles?

Thanks for your attention!

on the size ramsey number -...

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