Hypergraph containers
Andrew Thomason
David Saxton
Postgraduate Combinatorial Conference
Warwick 15th August 2012
Hypergraphs and containers
An r -uniform hypergraph G is a pair G = ([n], E ) where E ⊂ [n](r )
Hypergraphs and containers
An r -uniform hypergraph G is a pair G = ([n], E ) where E ⊂ [n](r )
A subset I ⊂ [n] is independent if there is no e ∈ E with e ⊂ I
Hypergraphs and containers
An r -uniform hypergraph G is a pair G = ([n], E ) where E ⊂ [n](r )
A subset I ⊂ [n] is independent if there is no e ∈ E with e ⊂ I
A good set of containers for G is a collection C ⊂ P[n] such that
• for each independent set I there exists C ∈ C with I ⊂ C
• each C ∈ C is not too big
• |C| is small
Hypergraphs and containers
An r -uniform hypergraph G is a pair G = ([n], E ) where E ⊂ [n](r )
A subset I ⊂ [n] is independent if there is no e ∈ E with e ⊂ I
A good set of containers for G is a collection C ⊂ P[n] such that
• for each independent set I there exists C ∈ C with I ⊂ C
• each C ∈ C is not too big
• |C| is small
Not good examples:
1. C = all independent sets
|C| is too big (maybe 2n/2 )
Hypergraphs and containers
An r -uniform hypergraph G is a pair G = ([n], E ) where E ⊂ [n](r )
A subset I ⊂ [n] is independent if there is no e ∈ E with e ⊂ I
A good set of containers for G is a collection C ⊂ P[n] such that
• for each independent set I there exists C ∈ C with I ⊂ C
• each C ∈ C is not too big
• |C| is small
Not good examples:
1. C = all independent sets
2. C = all maximal indpt sets
|C| is too big (maybe 2n/2 )
|C| is too big (maybe 2n/4 )
Hypergraphs and containers
An r -uniform hypergraph G is a pair G = ([n], E ) where E ⊂ [n](r )
A subset I ⊂ [n] is independent if there is no e ∈ E with e ⊂ I
A good set of containers for G is a collection C ⊂ P[n] such that
• for each independent set I there exists C ∈ C with I ⊂ C
• each C ∈ C is not too big
• |C| is small
Not good examples:
1. C = all independent sets
2. C = all maximal indpt sets
3. C = {[n]}
C ∈ C is big
|C| is too big (maybe 2n/2 )
|C| is too big (maybe 2n/4 )
Hypergraphs and containers
An r -uniform hypergraph G is a pair G = ([n], E ) where E ⊂ [n](r )
A subset I ⊂ [n] is independent if there is no e ∈ E with e ⊂ I
A good set of containers for G is a collection C ⊂ P[n] such that
• for each independent set I there exists C ∈ C with I ⊂ C
• each C ∈ C is not too big
• |C| is small
Not good examples:
1. C = all independent sets
2. C = all maximal indpt sets
3. C = {[n]}
|C| is too big (maybe 2n/2 )
|C| is too big (maybe 2n/4 )
C ∈ C is big
C ∈ C not too big will mean, say, |C | < 0.9n
1/(r −1)
|C| small will mean, say, |C | < 2n/d
d = average degree
Do they exist? Why do we care?
Example — list colourings
Let G be such that each v ∈ [n] has a list L(v ) of colours available
An L-colouring of G is a choice f (v ) ∈ L(v ) for each v ∈ [n]
so that f is a proper vertex colouring (no edge monochromatic)
The list-chromatic number χl (G ) is the smallest number ` so that,
provided ∀v |L(v )| ≥ `, there is always an L-colouring
(Vizing 1976, Erdős-Rubin-Taylor 1979)
Example — list colourings
Let G be such that each v ∈ [n] has a list L(v ) of colours available
An L-colouring of G is a choice f (v ) ∈ L(v ) for each v ∈ [n]
so that f is a proper vertex colouring (no edge monochromatic)
The list-chromatic number χl (G ) is the smallest number ` so that,
provided ∀v |L(v )| ≥ `, there is always an L-colouring
(Vizing 1976, Erdős-Rubin-Taylor 1979)
Clearly χl (G ) ≥ χ(G ) but can be much bigger
Eg χl (K3,3 ) = 3 χl (Kd,d ) = (1 + o(1)) log2 d
Example — list colourings
Let G be such that each v ∈ [n] has a list L(v ) of colours available
An L-colouring of G is a choice f (v ) ∈ L(v ) for each v ∈ [n]
so that f is a proper vertex colouring (no edge monochromatic)
The list-chromatic number χl (G ) is the smallest number ` so that,
provided ∀v |L(v )| ≥ `, there is always an L-colouring
(Vizing 1976, Erdős-Rubin-Taylor 1979)
Clearly χl (G ) ≥ χ(G ) but can be much bigger
Eg χl (K3,3 ) = 3 χl (Kd,d ) = (1 + o(1)) log2 d
Alon (1998,200): if G graph (r = 2) average degree d then
χl (G ) ≥ (1/2 + o(1)) log2 d
Example — list colourings
Let G be such that each v ∈ [n] has a list L(v ) of colours available
An L-colouring of G is a choice f (v ) ∈ L(v ) for each v ∈ [n]
so that f is a proper vertex colouring (no edge monochromatic)
The list-chromatic number χl (G ) is the smallest number ` so that,
provided ∀v |L(v )| ≥ `, there is always an L-colouring
(Vizing 1976, Erdős-Rubin-Taylor 1979)
Clearly χl (G ) ≥ χ(G ) but can be much bigger
Eg χl (K3,3 ) = 3 χl (Kd,d ) = (1 + o(1)) log2 d
Alon (1998,200): if G graph (r = 2) average degree d then
χl (G ) ≥ (1/2 + o(1)) log2 d
q
Haxell-Pei (2009): χl (G ) = Ω( logloglogd d ) for Steiner systems
Haxell-Verstraëte (2010): same for any simple 3-uniform
1
Alon-Kostochka (2010): Ω((log d) r −1 ) for simple r -uniform
List colourings and containers
Suppose there exists a set of containers C ⊂ P[n] with
(i) |C| ≤ e n/k and (ii) |C | ≤ (1 − c)n ∀C ∈ C
Then χl (G ) ≥ (1 + o(1)) log1/c k
List colourings and containers
Suppose there exists a set of containers C ⊂ P[n] with
(i) |C| ≤ e n/k and (ii) |C | ≤ (1 − c)n ∀C ∈ C
Then χl (G ) ≥ (1 + o(1)) log1/c k
Proof.
Pick random lists L(v ) size ` = (1 − ) log1/c k from t ≈ `2 cols
If ∃ colouring f then ∃ (C1 , . . . , Ct ) ∈ C t with ∀v v ∈ Cf (v )
List colourings and containers
Suppose there exists a set of containers C ⊂ P[n] with
(i) |C| ≤ e n/k and (ii) |C | ≤ (1 − c)n ∀C ∈ C
Then χl (G ) ≥ (1 + o(1)) log1/c k
Proof.
Pick random lists L(v ) size ` = (1 − ) log1/c k from t ≈ `2 cols
If ∃ colouring f then ∃ (C1 , . . . , Ct ) ∈ C t with ∀v v ∈ Cf (v )
Ie if B(v ) = {i ∈ [t] : v ∈ Ci } then f (v ) ∈ B(v ) ∩ L(v ) 6= ∅
List colourings and containers
Suppose there exists a set of containers C ⊂ P[n] with
(i) |C| ≤ e n/k and (ii) |C | ≤ (1 − c)n ∀C ∈ C
Then χl (G ) ≥ (1 + o(1)) log1/c k
Proof.
Pick random lists L(v ) size ` = (1 − ) log1/c k from t ≈ `2 cols
If ∃ colouring f then ∃ (C1 , . . . , Ct ) ∈ C t with ∀v v ∈ Cf (v )
Ie if B(v ) = {i ∈ [t] : v P
∈ Ci } thenPf (v ) ∈ B(vP
) ∩ L(v ) 6= ∅
Let |B(v )| = bv t; then v bv t = |B(v )| = i |Ci | ≤ t(1 − c)n
List colourings and containers
Suppose there exists a set of containers C ⊂ P[n] with
(i) |C| ≤ e n/k and (ii) |C | ≤ (1 − c)n ∀C ∈ C
Then χl (G ) ≥ (1 + o(1)) log1/c k
Proof.
Pick random lists L(v ) size ` = (1 − ) log1/c k from t ≈ `2 cols
If ∃ colouring f then ∃ (C1 , . . . , Ct ) ∈ C t with ∀v v ∈ Cf (v )
Ie if B(v ) = {i ∈ [t] : v P
∈ Ci } thenPf (v ) ∈ B(vP
) ∩ L(v ) 6= ∅
Let |B(v )| = bv t; then v bv t = |B(v )| = i |Ci | ≤ t(1 − c)n
t −1
v )t
Let pv = Pr{B(v ) ∩ L(v ) = ∅} = (1−b
`
`
List colourings and containers
Suppose there exists a set of containers C ⊂ P[n] with
(i) |C| ≤ e n/k and (ii) |C | ≤ (1 − c)n ∀C ∈ C
Then χl (G ) ≥ (1 + o(1)) log1/c k
Proof.
Pick random lists L(v ) size ` = (1 − ) log1/c k from t ≈ `2 cols
If ∃ colouring f then ∃ (C1 , . . . , Ct ) ∈ C t with ∀v v ∈ Cf (v )
Ie if B(v ) = {i ∈ [t] : v P
∈ Ci } thenPf (v ) ∈ B(vP
) ∩ L(v ) 6= ∅
Let |B(v )| = bv t; then v bv t = |B(v )| = i |Ci | ≤ t(1 − c)n
t −1
v )t
Let pv = Pr{B(v ) ∩ L(v ) = ∅} = (1−b
`
`
−1
P
≈ nc ` = nk −1+
so v pv ≥ n ct` t`
List colourings and containers
Suppose there exists a set of containers C ⊂ P[n] with
(i) |C| ≤ e n/k and (ii) |C | ≤ (1 − c)n ∀C ∈ C
Then χl (G ) ≥ (1 + o(1)) log1/c k
Proof.
Pick random lists L(v ) size ` = (1 − ) log1/c k from t ≈ `2 cols
If ∃ colouring f then ∃ (C1 , . . . , Ct ) ∈ C t with ∀v v ∈ Cf (v )
Ie if B(v ) = {i ∈ [t] : v P
∈ Ci } thenPf (v ) ∈ B(vP
) ∩ L(v ) 6= ∅
Let |B(v )| = bv t; then v bv t = |B(v )| = i |Ci | ≤ t(1 − c)n
t −1
v )t
Let pv = Pr{B(v ) ∩ L(v ) = ∅} = (1−b
`
`
−1
P
≈ nc ` = nk −1+
so v pv ≥ n ct` t`
P
Q
−1+
so Pr{Ci ’s ok} = v (1 − pv ) ≤ e − v pv ≤ e −nk
List colourings and containers
Suppose there exists a set of containers C ⊂ P[n] with
(i) |C| ≤ e n/k and (ii) |C | ≤ (1 − c)n ∀C ∈ C
Then χl (G ) ≥ (1 + o(1)) log1/c k
Proof.
Pick random lists L(v ) size ` = (1 − ) log1/c k from t ≈ `2 cols
If ∃ colouring f then ∃ (C1 , . . . , Ct ) ∈ C t with ∀v v ∈ Cf (v )
Ie if B(v ) = {i ∈ [t] : v P
∈ Ci } thenPf (v ) ∈ B(vP
) ∩ L(v ) 6= ∅
Let |B(v )| = bv t; then v bv t = |B(v )| = i |Ci | ≤ t(1 − c)n
t −1
v )t
Let pv = Pr{B(v ) ∩ L(v ) = ∅} = (1−b
`
`
−1
P
≈ nc ` = nk −1+
so v pv ≥ n ct` t`
P
Q
−1+
so Pr{Ci ’s ok} = v (1 − pv ) ≤ e − v pv ≤ e −nk
so Pr{∃f } ≤ |C|t e −nk
−1+
≤ e (log
2
k)nk −1 e −nk −1+
< 1.
List colouring results for simple r -uniform
Saxton+T (2011) by randomized container construction:
1
χl (G ) ≥ 2r log(2r
2 ) + o(1) log d for d-regular
Saxton+T (2012) by deterministic container construction:
1
χl (G ) ≥ (1 + o(1)) (r −1)
2 logr d for average degree d
1
χl (G ) ≥ (1 + o(1)) r −1
logr d for d-regular
average degree case needs better machinery
H-free hypergraphs
Let H be an `-uniform hypergraph. We say G is H-free if H is not
a subgraph of G .
−1
Let π(H) = limn→∞ max{e(G ) : |G | = n, G is H-free} n`
H-free hypergraphs
Let H be an `-uniform hypergraph. We say G is H-free if H is not
a subgraph of G .
−1
Let π(H) = limn→∞ max{e(G ) : |G | = n, G is H-free} n`
Classical stuff:
for r = 2 π(H) = 1 −
1
χ(H)−1
(Turán,Erdős-Stone, Simonovits)
n
for r = 2 # H-free graphs = 2(1+o(1))π(H)(2)
(Erdős,Kleitman,Rothschild,Frankl,Rödl)
H-free hypergraphs
Let H be an `-uniform hypergraph. We say G is H-free if H is not
a subgraph of G .
−1
Let π(H) = limn→∞ max{e(G ) : |G | = n, G is H-free} n`
Classical stuff:
for r = 2 π(H) = 1 −
1
χ(H)−1
(Turán,Erdős-Stone, Simonovits)
n
for r = 2 # H-free graphs = 2(1+o(1))π(H)(2)
(Erdős,Kleitman,Rothschild,Frankl,Rödl)
for r ≥ 3 π(H) unknown
n
for r ≥ 3 # H-free graphs = 2(1+o(1))π(H)(`)
(Nagle,Rödl,Schacht)
Sparse Turán
Theorem
Let G (`) (n, p) be a random `-uniform hypergraph. Then, almost
surely, no H-free subgraph of G has more than
n
(π(H) + )p
`
edges, provided p > Cn−1/m(H) .
Here m(H) = maxH 0 ⊂H
e(H 0 )−1
.
v (H 0 )−`
Sparse Turán
Theorem
Let G (`) (n, p) be a random `-uniform hypergraph. Then, almost
surely, no H-free subgraph of G has more than
n
(π(H) + )p
`
edges, provided p > Cn−1/m(H) .
Here m(H) = maxH 0 ⊂H
e(H 0 )−1
.
v (H 0 )−`
Conjectured by Kohayakawa-Luczak-Rödl (1997)
Proved by Conlon-Gowers (2010, strictly balanced), Schacht (2010)
Containers and H-free graphs
Let H be an
`-uniform graph.
Let n = N` . Let r = e(H).
The r -uniform hypergraph G (N, H) has n vertices = [N](`) .
The edges are subsets of V (G ) spanning a copy of H in [N].
Containers and H-free graphs
Let H be an
`-uniform graph.
Let n = N` . Let r = e(H).
The r -uniform hypergraph G (N, H) has n vertices = [N](`) .
The edges are subsets of V (G ) spanning a copy of H in [N].
The subsets of V (G ) are `-graphs on vertex set [N].
The independent subsets of V (G ) are H-free `-graphs.
Containers and H-free graphs
Let H be an
`-uniform graph.
Let n = N` . Let r = e(H).
The r -uniform hypergraph G (N, H) has n vertices = [N](`) .
The edges are subsets of V (G ) spanning a copy of H in [N].
The subsets of V (G ) are `-graphs on vertex set [N].
The independent subsets of V (G ) are H-free `-graphs.
A good set of containers for G is, therefore, a small collection of
`-graphs on vertex set [N], containing between them all H-free
`-graphs, and no graph in the collection is “large”
A theorem on `-graphs
Theorem (Saxton+T)
Let H be an `-graph with e(H) ≥ 2 and let > 0. For some c > 0
and for N sufficiently large, there exists a collection C of `-graphs
on vertex set [N] such that
(a) for every H-free `-graph I on vertex set [N] there exists C ∈ C
with I ⊂ C ,
(b) for every `-graph C ∈ C, the number of copies of H in C is at
most N v (H) , and e(C ) ≤ (π(H) + ) N` ,
(c) log |C| ≤ cN `−1/m(H) log N .
A theorem on `-graphs
Theorem (Saxton+T)
Let H be an `-graph with e(H) ≥ 2 and let > 0. For some c > 0
and for N sufficiently large, there exists a collection C of `-graphs
on vertex set [N] such that
(a) for every H-free `-graph I on vertex set [N] there exists C ∈ C
with I ⊂ C ,
(b) for every `-graph C ∈ C, the number of copies of H in C is at
most N v (H) , and e(C ) ≤ (π(H) + ) N` ,
(c) log |C| ≤ cN `−1/m(H) log N .
Proof of Sparse Turán: Given C ∈ C, Pr{G (`) (N, p) has ≥ (π(H) + 2)p N` edges of C } < exp{−pN ` }
So Pr{this happens for any C } ≤ |C| exp{−pN ` } = o(1)
Further results via containers
Kohayakawa-Luczak-Rödl made 3 conjectures, one of which was
outstanding (“the KLR conjecture”) but all follow easily from the
theorem. The KLR conjecture for balanced graphs was proved by
Balogh-Morris-Samotij (2012) in a closely related way.
Further results via containers
Kohayakawa-Luczak-Rödl made 3 conjectures, one of which was
outstanding (“the KLR conjecture”) but all follow easily from the
theorem. The KLR conjecture for balanced graphs was proved by
Balogh-Morris-Samotij (2012) in a closely related way.
Similar results for solution to linear equations can be proved. Eg
Theorem (Conlon and Gowers, Schacht)
Let ` ≥ 3 and > 0. There exists c > 0 such that for
p ≥ cN −1/(`−1) , if X ⊂ [N] is a random subset chosen with
probability p, then almost surely, any subset of X of size |X |
contains an arithmetic progression of length `.
Theorem (Saxton+T)
√
√
There are between 2(1.16+o(1)) n and 2(55+o(1)) n Sidon subsets
of [n]. (These are sets with no non-trivial solution to a + b = c + d.)
OK — how about finding some containers?
This simple algorithm for d-regular graphs is due to Sapozhenko.
We assume V (G ) = [n]. For S ⊂ V (G ) let Γ(S) be its neighbours.
Let I ⊂ V (G ). Let ζ > 0. Start with T = ∅.
for v = 1, 2, . . . , n do:
if v ∈ I and |Γ(T ∪ {v })| ≥ |Γ(T )| + ζd then add v to T
OK — how about finding some containers?
This simple algorithm for d-regular graphs is due to Sapozhenko.
We assume V (G ) = [n]. For S ⊂ V (G ) let Γ(S) be its neighbours.
Let I ⊂ V (G ). Let ζ > 0. Start with T = ∅.
for v = 1, 2, . . . , n do:
if v ∈ I and |Γ(T ∪ {v })| ≥ |Γ(T )| + ζd then add v to T
Note afterwards |T | ≤ n/ζd.
OK — how about finding some containers?
This simple algorithm for d-regular graphs is due to Sapozhenko.
We assume V (G ) = [n]. For S ⊂ V (G ) let Γ(S) be its neighbours.
Let I ⊂ V (G ). Let ζ > 0. Start with T = ∅.
for v = 1, 2, . . . , n do:
if v ∈ I and |Γ(T ∪ {v })| ≥ |Γ(T )| + ζd then add v to T
Note afterwards |T | ≤ n/ζd.
Let T ⊂ V (G ). Let ζ > 0. Start with S = ∅, C = [n] − Γ(T ).
for v = 1, 2, . . . , n do:
if |Γ(S ∪ {v })| ≥ |Γ(S)| + ζd then
if v ∈ T add v to S else take v from C
OK — how about finding some containers?
This simple algorithm for d-regular graphs is due to Sapozhenko.
We assume V (G ) = [n]. For S ⊂ V (G ) let Γ(S) be its neighbours.
Let I ⊂ V (G ). Let ζ > 0. Start with T = ∅.
for v = 1, 2, . . . , n do:
if v ∈ I and |Γ(T ∪ {v })| ≥ |Γ(T )| + ζd then add v to T
Note afterwards |T | ≤ n/ζd.
Let T ⊂ V (G ). Let ζ > 0. Start with S = ∅, C = [n] − Γ(T ).
for v = 1, 2, . . . , n do:
if |Γ(S ∪ {v })| ≥ |Γ(S)| + ζd then
if v ∈ T add v to S else take v from C
So T specifies C , and if T came from I then I ⊂ C .
OK — how about finding some containers?
This simple algorithm for d-regular graphs is due to Sapozhenko.
We assume V (G ) = [n]. For S ⊂ V (G ) let Γ(S) be its neighbours.
Let I ⊂ V (G ). Let ζ > 0. Start with T = ∅.
for v = 1, 2, . . . , n do:
if v ∈ I and |Γ(T ∪ {v })| ≥ |Γ(T )| + ζd then add v to T
Note afterwards |T | ≤ n/ζd.
Let T ⊂ V (G ). Let ζ > 0. Start with S = ∅, C = [n] − Γ(T ).
for v = 1, 2, . . . , n do:
if |Γ(S ∪ {v })| ≥ |Γ(S)| + ζd then
if v ∈ T add v to S else take v from C
So T specifies C , andif T came from I then I ⊂ C .
n
There are at most ζd
≤ e n/ζd sets C .
OK — how about finding some containers?
This simple algorithm for d-regular graphs is due to Sapozhenko.
We assume V (G ) = [n]. For S ⊂ V (G ) let Γ(S) be its neighbours.
Let I ⊂ V (G ). Let ζ > 0. Start with T = ∅.
for v = 1, 2, . . . , n do:
if v ∈ I and |Γ(T ∪ {v })| ≥ |Γ(T )| + ζd then add v to T
Note afterwards |T | ≤ n/ζd.
Let T ⊂ V (G ). Let ζ > 0. Start with S = ∅, C = [n] − Γ(T ).
for v = 1, 2, . . . , n do:
if |Γ(S ∪ {v })| ≥ |Γ(S)| + ζd then
if v ∈ T add v to S else take v from C
So T specifies C , andif T came from I then I ⊂ C .
n
There are at most ζd
≤ e n/ζd sets C .
C − T sends ≥ (1 − ζ)d|C − T | edges into Γ(T ) so |C | ≤
n
2−ζ
n
+ ζd
What about r -uniform hypergraphs?
We give an algorithm which operates in two modes.
In build mode it constructs C (Tr −1 , . . . , T0 ) ⊂ V (G ) from sets
Tr −1 , . . . , T0 ⊂ V (G ).
In prune mode, given I ⊂ V (G ), it finds small sets
Tr −1 , . . . , T0 ⊂ I so that I ⊂ C (Tr −1 , . . . , T0 ).
In each mode it constructs a sequence of s-uniform hypergraphs
Ps , s = r , r − 1, . . . , 1.
Each edge of Ps is an edge of Ps+1 whose first vertex, which is
in Ts , has been removed.
During the construction of Ps , a hypergraph Γs of subsets that
have reached a high degree is maintained.
Algorithm — description
Algorithm
input an r -graph G on vertex set [n]
an (s + 1)-multigraph Ps+1 on vertex set [n]
parameters τ, ζ > 0
in prune mode a subset I ⊂ [n]
in build mode a subset Ts ⊂ [n]
output an s-multigraph Ps on vertex set [n]
in prune mode a subset Ts ⊂ [n]
in build mode a subset Cs ⊂ [n]
Algorithm — running
put E (Ps ) = ∅ and Γs = ∅
in prune mode put Ts = ∅
in build mode put Cs = [n]
for v = 1, 2, . . . , n do:
let F = {f ∈ [v + 1, n](s) : {v } ∪ f ∈ E (Ps+1 ), and ∀σ ∈ Γs σ 6⊂ f }
[here F is a multiset with multiplicities inherited from E (Ps+1 )]
in prune mode if |F | ≥ ζτ r −s−1 d(v ) and v ∈ I , add v to Ts
in build mode
if |F | ≥ ζτ r −s−1 d(v ), remove v from Cs
if v ∈ Ts then
add F to E (Ps )
for each u ∈ [v + 1, n],
if ds (u) > τ r −s d(u), add {u} to Γs
(>1)
for each σ ∈ [v + 1, n]
, if ds (σ) > 2s τ ds+1 (σ), add σ to Γs