Further applications of the Container Method

0.16 MB·

by  Jozsef Balogh

Further applications of the Container Method
Jo´zsefBaloghandAdamZs.Wagner
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AbstractRecently,Balogh–Morris–SamotijandSaxton–Thomasonprovedthathy-

pergraphssatisfyingsomenaturalconditionshaveonlyfewindependentsets.Their
]
O main results already have several applications. However, the methods of proving
C these theorems are even more far reaching. The general idea is to describe some
. familyofevents,whosecardinalityaprioricouldbelarge,onlywithafewcertifi-
h
cates.Here,weshowsomeapplicationsofthemethods,includingcountingC -free
t 4
a graphs, considering the size of a maximumC -free subgraph of a random graph
4
m
andcountingmetricspaceswithagivennumberofpoints.Additionally,wediscuss
[ someconnectionswiththeSzemere´diRegularityLemma.

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1 Introduction
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Recently, an importanttrend in probabilistic combinatoricsis to transfer extremal
0
results in a dense environmentinto a sparse, random environment.The first main
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breakthrough is due to Conlon–Gowers [8] and Schacht [23]. Not much later,
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Balogh–Morris–Samotij[2] and Saxton–Thomason[22] had a differentapproach,
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1 whichnotonlyprovedmostoftheresultsofConlon–Gowers[8]andSchacht[23],
: but also provided counting versions of these results. Additionally, there are some
v
furtherapplicationsofthemaintheoremsof[2]and[22].However,theproofmeth-
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r
a Jo´zsefBalogh
DepartmentofMathematicalSciences,UniversityofIllinoisatUrbana-Champaign,Urbana,Illi-
nois61801,USAe-mail:[email protected]. ResearchispartiallysupportedbySimons
Fellowship, NSA Grant H98230-15-1-0002, NSF CAREER Grant DMS-0745185, NSF Grant
DMS-1500121,ArnoldO.BeckmanResearchAward(UIUCCampusResearchBoard15006)and
MarieCurieFP7-PEOPLE-2012-IIF327763.
AdamZs.Wagner
DepartmentofMathematicalSciences,UniversityofIllinoisatUrbana-Champaign,Urbana,Illi-
nois61801,USAe-mail:[email protected].
1
2 Jo´zsefBaloghandAdamZs.Wagner
odshavethepotentialtobemoreinfluentialthanthetheoremsthemselves.Thegen-
eralidea,whichcanbetracedbacktotheclassicalpaperofKleitman–Winston[13]
(andmoreexplicitlyinseveralpapersofSapozhenko)istodescribesomefamilyof
events,whosecardinalityaprioricouldbelarge,onlywithafewcertificates.
Here,wetrytooutlinewhatliesbehindthismethod.Becauseeachoftheabove
mentionedfourpapers[8, 23, 2, 22] already gavea nice surveyof the field, addi-
tionallyConlon[7],Ro¨dl–Schacht[20]andSamotij[21]havesurveypapersofthe
topics, we choose a different route. Each of the following four sections discusses
themethodwithsomeapplications.Threeofthesectionscontainnewresults.
InSection2 we state an importantcorollaryof themainresultsof [2]and [22]
andwilldiscussitsconnectionwiththeSzemere´diRegularityLemma[24].
(n)
In Section 3 we estimate the volumeof the convexsubsetof [0,1] 2 represen-
tingmetricspaceswithnpoints,andconsideradiscretevariantofthisproblemas
well. Kozma–Meyerovitch–Peled–Samotij[15] consideredthe followingquestion:
chooserandomlyandindependently n numbersfromtheinterval[0,1],labelling
2
the edges of a complete graph K , what is the probability that any three of them
n
(cid:0) (cid:1)
formingatrianglewillsatisfythetriangleinequality?Theyusedentropyestimates
toderiveanupperboundonthisprobabilityandnoticedthatSzemere´diRegularity
Lemmacanbeusedtocountmetricspaceswhosealldistancesbelongtoadiscrete
setofafixedsize.
Mubayi and Terry [19] more recently pushed the discrete case into 0,1 -law
{ }
typeresults.Usingthegeneralresultsof[2]and[22],weimprovetheresultsimplied
bytheregularitylemma,andobtaingoodresultsforthecontinuouscase.Parallelto
ourwork,Kozma,Meyerovitch,Morris,Peled andSamotij[16] practicallysolved
thecontinuousversionoftheproblemusingmoreadvancedversionsofthemethods
of[2]and[22].Inordertoavoidduplicatework,ourgoalinthissectionisclarity
overpushingthemethodtoitslimit.
It was probably Babai–Simonovits–Spencer [1] who first considered extremal
problemsinrandomgraphs.Mostoftheirproposedproblemsareresolved,butthere
wasnoprogressonthefollowing:Whatisthemaximumnumberoftheedgesofa
C -freesubgraphoftherandomgraphG(n,p)when p=1/2?Inthecase p=o(1),
4
strongboundsweregivenbyKohayakawa–Kreuter–Steger[14].Here,inSection4,
we improve on the trivial upper bound, noting that an n-vertexC -free graph can
4
haveatmost(1/2+o(1))n3/2edges.
Theorem1.1 For every p (0,1), there is a c>0 that the largestC -free
4
∈
subgraph of G(n,p) has at most (1/2 c)n3/2 edges w.h.p. In particular, if
−
p=0.5wecantakec=0.028.
Kleitman–Winston[13]authoredoneofthefirstpapersinthefieldwhosemain
ideawastofindsmallcertificatesoffamiliesofsetsinordertoprovethatthereare
notmanyofthem.TheyprovedthatthenumberofC -freegraphsonnverticesisat
4
most2cn3/2 forc 1.081919.InSection5,weimprovetheconstantintheexponent
≈
FurtherapplicationsoftheContainerMethod 3
by using ideas found in Section 3. The improvement is tiny; the main aim is to
demonstratethatthemethodhaspotentialapplicationsforotherproblemsaswell.
2 Connections to Szemere´di’s RegularityLemma
In this section we describe some connections between the Szemere´di Regularity
Lemma and the new counting method. Originally, without using the regularity
lemma,Erdo˝s–Kleitman–Rothschild[10]estimatedthenumberofK -freegraphs.
k
Theorem2.1 Thereare2(1+o(1))·ex(n,Kk) Kk-freegraphsonnvertices,where
ex(n,K )isthemaximumnumberoftheedgesofaK -freegraphonnvertices.
k k
Here, we sketch a well-knownproofof Erdo˝s–Frankl–Ro¨dl[9], using the Sze-
mere´diRegularityLemma.WeapplytheRegularityLemmaforaK -freegraphG ,
k n
which outputs a cluster graph R, wheret =O(1). Then we clean G , i.e., we re-
t n
move edges inside the clusters, between the sparse and the irregularpairs. Define
C to be the blow-up of R to n vertices(getting back the same preimage vertices
n t
ofG ).ObservethatC containsallbuto(n2)edgesofG andC isK -free,hence
n n n n k
e(C ) ex(n,K ).NowwecancountthenumberofchoicesforG :Thenumberof
n k n
≤
choices forC is O(1) nn, the numberof choices for E(G ) E(C ) givenC , is
n n n n
· ∩
atmost2ex(n,Kk),andthenumberofchoicesforE(Gn) E(Cn)givenCn,is2o(n2),
−
completingtheproofoftheresult.
An important corollary of the main results of Balogh–Morris–Samotij [2] and
Saxton–Thomason[22]isthefollowingcharacterizatonofK -freegraphs.
k
Proposition2.2 There is a t 2O(logn·n2−1/(k−1)) and a set G1,...,Gt of
≤ { }
graphs,eachcontainingo(nk)copiesofK ,suchthatforeveryK -freegraph
k k
H thereisani [t]suchthatH G.
i
∈ ⊆
Notethatitfollowsfromstandardresultsinextremalgraphtheorythateachsuch
graphG necessarilycontainsatmost(1+o(1)) ex(n,K )edges.
i k
·
Thisprovidesan evenshorterproofofTheorem2.1:Foreach K -freegraphH
k
thereisani [t]suchthatH Gi.Thenumberofchoicesforiis2O(logn·n2−1/(k−1)),
andthenum∈berofsubgraphs⊆ofGiisatmost2(1+o(1))ex(n,Kk).
TheproofusingtheRegularityLemmayieldstwovariantsofProposition2.2,one
whichisweaker,asitgivest=2o(n2),thoughitisstrongenoughforthisparticular
application,andonethatwecalltheSzemere´diApproximateContainerLemma.
4 Jo´zsefBaloghandAdamZs.Wagner
Proposition2.3 Thereisat =2o(n2) andaset G ,...,G ofgraphs,each
1 t
containing at most (1+o(1))ex(n,K ) edges, a{nd o(nk) co}pies of K , such
k k
thatforeveryK -freegraphH thereisani [t]suchthatH G.
k i
∈ ⊆
Proposition2.4 There isat =O(1)anda set G ,...,G ofgraphs,each
1 t
{ }
K -free,suchthatforeveryK -free graphH there isani [t]andapermu-
k k
∈
tationofV(H),givinganisomorphicgraphH ,suchthat E(H ) E(G) =
′ ′ i
| − |
o(n2).
Most of the tools fromthis section are likely to be usefulin countingmaximal
K -freegraphs,wheretherearestillnosatisfactoryboundswhenr 4.
r
≥
Problem2.5 WhatisthenumberofmaximalK -freegraphswithvertexset
r
[n]?Forr=3thiswasaquestionofErdo˝s,whichwassettledin[4]and[3].
3 The number ofmetricspaces
Ourgoalistoestimatethenumberofmetricspacesonnpoints,wherethedistance
betweenanytwopointsliesin 1,...,r forsomer=r(n).Thisproblemwascon-
{ }
sideredfirstbyKozma–Meyerovitch–Peled–Samotij[15],who,usingtheregularity
lemmagaveanasymptoticboundonthenumberofsuchmetricspacesfora fixed
constantr. Recently,Mubayi–Terry[19] provideda characterisationofthe typical
structureofsuchmetricspacesforafixedconstantr,whilen ¥ .Wewillbemore
→
interestedinwhathappensifrisallowedtogrowasafunctionofn.Ourmainresult
isthefollowing:
Theorem3.1 Fixanarbitrarysmallconstante >0.If
n1/3
r=O ,
log43+e n!
thenthenumberofsuchmetricspacesis
FurtherapplicationsoftheContainerMethod 5
r+1 (n2)+o(n2)
.
2
(cid:24) (cid:25)
Kozma–Meyerovitch–Peled–Samotij[15] pointed out that the discrete and the
continuousproblemsarerelated.Theyconsideredthesamequestioninthecontinu-
ouscasewithdistancesin[0,1].TheirentropybasedapproachyieldsTheorem3.1
forr<n1/8.
At the end of this section we show how our results translate to the continuous
setting.
For a positive integer r define m(r)= r+1 . We will use an easy corollary of
⌈ 2 ⌉
Mubayi–Terry([19]Lemma4.9):
Lemma3.2 LetA,B,C [r], allnon-empty.Supposethetriple A,B,C doesnot
⊂ { }
contain a non-metric triangle – that is, every triple a,b,c:a A,b B,c C
{ ∈ ∈ ∈ }
satisfies the triangle-inequality.Then if r is evenwe have A + B + C 3m(r),
| | | | | |≤
andifrisoddwehave A + B + C 3m(r)+1.
| | | | | |≤
LetH bethe3-uniformhypergraphwithvertexsetr rows,oneforeachcolor,
and n columns, one for each edge of K . A vertex (i,f) of H corresponds to
2 n
theeventthatthegraphedge f hascolori.ThreeverticesofH formahyperedge
(cid:0) (cid:1)
when the graph edge coordinates of the vertices form a triangle in K while the
n
‘colors’ do not satisfy the triangle inequality. With other words, the hyperedges
correspondtonon-metrictriangles,andindependentsetshavingexactlyonevertex
fromeachcolumncorrespondtopointsofthemetricpolytope.Ourplanistoprove
asupersaturationstatement,butfirstweneedtwolemmas.
The first lemma we use is due to Fu¨redi [11]. For a graph G, write G2 for the
“propersquare”ofG,i.e.,wherexyisanedgeifandonlyifthereisazsuchthatxz
andzyareedgesinG.Writee(G)forthenumberofedgesinG.
Lemma3.3 ForanygraphGwithnvertices,wehave
e(G2) e(G) n/2 .
≥ −⌊ ⌋
ThesecondlemmaboundsthesizeofthelargestindependentsetinH.
Lemma3.4 Let S V(H) have no empty columnsand containno edges in H.
Thenifrisevenwe⊂have S m(r) n ,andifrisoddwehave S m(r) n +rn.
| |≤ 2 | |≤ 2
Proof. Theevencasefollowsdirectl(cid:0)y(cid:1)fromLemma3.2,andwenotethatth(cid:0)is(cid:1)bound
istight–letScontaintheinterval[r/2,r]fromeachcolumn.Theboundintheodd
case is slightly more difficult, and we make no effort to establish a tight bound,
whichshouldprobablybe S m(r) n +n/2.
| |≤ 2
Letrbeodd,andletA,B,CbethreecolumnsthatformatriangleofS.Notethatif
(cid:0) (cid:1)
forsomek 1wehave A m(r)+kand B m(r)+kthen C m(r) 2k+1
≥ | |≥ | |≥ | |≤ − ≤
m(r) kbyLemma3.2.WriteB forthesetofcolumnsinSoforderatleastm(r)+k
k
−
6 Jo´zsefBaloghandAdamZs.Wagner
andwriteS forthesetofcolumnsinSoforderatmostm(r) k.LetG bethegraph
k k
−
on[n]withedgesB .ThenbyLemma3.3weget
k
S e(G2) e(G ) n/2 B n.
| k|≥ k ≥ k −⌊ ⌋≥| k|−
Hence
n (cid:229) r (cid:229) r (cid:229) r
S m(r) = k(B B ) k(S S )= (B S ) nr,
| |− 2 | k|−| k+1| − | k|−| k+1| | k|−| k| ≤
(cid:18) (cid:19) k=1 k=1 k=1
andtheresultfollows.
⊓⊔
Nowwearereadytoproveasupersaturation-likeresult.
Lemma3.5 Lete >0andletS V(H)withnoemptycolumns.
1. Ifrisevenand S (1+e )⊂n m(r),thenScontainsatleast e n hyperedges.
| |≥ 2 10 3
2. Ifr is odd,n>ne sufficientlylargeand S (1+e ) n m(r), thenScontains
(cid:0) (cid:1) | |≥ 2 (cid:0) (cid:1)
atleast e4 n hyperedges.
40000 3 (cid:0) (cid:1)
Proof. Supposefi(cid:0)rs(cid:1)tthatriseven.Thenthereareatleast e n trianglesinGsuch
10 3
that the correspondingcolumns contain at least (1+e /10)3m(r) vertices from S.
(cid:0) (cid:1)
Indeed,ifthiswasnotthecase,then
e n 3r+ n 3m(r)(1+e /10) n e (r+m(r))
10 3 3 = m(r)+ > S,
n 2 2 10 | |
(cid:0) (cid:1) (cid:0) (cid:1)− (cid:18) (cid:19)(cid:18) (cid:19)
whichisacontradiction.Hencepart1ofthelemmafollowsfromLemma3.2.
Now suppose r is odd. Given T [n], write f (T) for the set of vertices of S
S
⊂
containedinthe T columnsofH correspondingtotheedgesspannedbyT.Set
|2|
n =20/e , so thatby Lemma3.4, wheneverT [n] with T =n and f (T)
0 0 S
m(r) n0 (1+e )(cid:0)then(cid:1)H[f (T)]containsahyper⊂edge. | | | |≥
2 3 S
First,weclaimthatthereareatleast e n choicesofT [n]with T =n and
f (T(cid:0)) (cid:1) m(r) n0 (1+e ). 4 n0 ⊂ | | 0
| S |≥ 2 3 (cid:0) (cid:1)
Indeed,ifthiswasnotthecase,thenwewouldhave
(cid:0) (cid:1)
e n n0 r+ n m(r) n0 (1+e ) n e r e m(r)
S 4 n0 2 n0 2 3 = m(r)+ +
| |≤ (cid:0) (cid:1)(cid:0) (cid:1) (cid:0)nn0−(cid:1)−22 (cid:0) (cid:1) (cid:18)2(cid:19)(cid:18) 4 3 (cid:19) (1)
< n m(r)(1+(cid:0)e ), (cid:1)
2
(cid:18) (cid:19)
whichisnotpossible.SothenumberofhyperedgescontainedinSisatleast
e n n 3 e e 3 n
e(H[S]) / − ,
≥ 4 n n 3 ≥ 4203 3
(cid:18) 0(cid:19) (cid:18) 0− (cid:19) (cid:18) (cid:19)
andtheresultfollows.
⊓⊔
FurtherapplicationsoftheContainerMethod 7
Write d¯fortheaveragedegreeof H, andfor j [3] definethe j-th maximum
∈
co-degree
D =max e E(H):s e :s V(H)and s = j .
j
{|{ ∈ ⊂ }| ⊂ | | }
Below,wewillmakeuseofaversionofthecontainertheoremof[2,22],thewayit
wasformulatedbyMousset–Nenadov–Steger[18].
Theorem3.6 There exists a positive integerc such thatthe following holds
for every positive integer N. Let H be a 3-uniform hypergraphof order N.
Let0 p 1/(36c)and0<a <1besuchthatD (H,p) a /(27c),where
≤ ≤ ≤
4D 2D
D (H ,p)= 2 + 3.
d¯p d¯p2
ThenthereexistsacollectionofcontainersC P(V(H))suchthat
(i)everyindependentsetinH iscontained⊂insomeC C,
(ii)forallC C wehavee(H[C]) a e(H ),and ∈
∈ ≤
(iii)thenumberofcontainerssatisfies
log C 39c(1+log(1/a ))Nplog(1/p).
| |≤
Proof of Theorem 3.1. Let H be the hypergraph defined earlier, i.e., the 3-
uniform hypergraph with vertex set formed by pairs of the r colors and the n
2
edgesofK ,with3-edgescorrespondingtonon-metrictriangles.Lete ,d >0bear-
n
bitrarilysmallconstantsandset p= 1 anda = 1010clog4+2d n.InH weha(cid:0)v(cid:1)e
rlog2+d n n
D nr2, D r, D =1, d¯ r2n/64and
1 2 3
≤ ≤ ≥
64r2log2+d n 64r2log4+2d n a
D (H,p) 4 + .
≤ r2n 2r2n !≤ 27c
ThenTheorem3.6providescontainerswith
e(H[C]) a e(H) 104cr3n2log4+2d n,
≤ ≤
andthenumberofcontainersis
c310rn2 logn logr loglogn
log C · · · =o(n2).
| |≤ rlog2+d n
Nowassume
n1/3
r=o .
log(4+2d )/3n!
8 Jo´zsefBaloghandAdamZs.Wagner
Thenthemaximumnumberofedgesinacontaineriso(n3),hencebyLemma3.5,
andthefactthatausefulcontainerdoesnothaveanemptycolumn,wehaveforn
largeenough,
n
V(C) <(1+e )m(r) .
| | 2
(cid:18) (cid:19)
Hence, the number of colourings in a container is at most (1+e )(n2)m(r)(n2) =
m(r)(n2)+o(n2).Thelogarithmofthenumberofcontainersiso(n2).Thetotalnumber
ofgoodcolouringsisatmostthenumberofcontainerstimesthemaximumnumber
ofcolouringsinacontainer.Hencethetotalnumberofgoodcolouringsis
m(r)(n2)+o(n2),
asrequired.
⊓⊔
Now we turn our attention to the continuous setting. The set-up in [15] is as
follows. Given a metric space with n points and all distances being in [0,1], we
(n)
regardthesetofdistancesasavectorin[0,1] 2 .Wewillcalltheunionofallsuchn
(n)
pointsin[0,1] 2 forallfinitemetricspacesthemetricpolytopeMn.Moreprecisely,
the metric polytopeMn is the convexpolytopein R(n2) defined by the inequalities
0<d 1andd d +d .
ij ij ik jk
≤ ≤
Notethatifa+b cthen
≥
a + b c . (2)
⌈ ⌉ ⌈ ⌉≥⌈ ⌉
Theorem3.7 Fixd >0constant.Thenforn>nd sufficientlylarge,wehave
(vol(Mn))1/(n2)≤ 21+n611−d .
Proof. First consider the discrete setting, colouring with r colours, where r is the
evenintegerclosestton16−d2.W.l.o.g.d <1/4andset
1/p=n31−d4, a =300cnd −32,
wherecistheconstantfromTheorem3.6.Then
1 1 n1/3 d /4 n2/3 d /2
D (H,p)<300 + 300 − + − a ,
(cid:18)rnp p2r2n(cid:19)≤ n7/6−d /2 n4/3−d !≤
andwegetcontainerswith
e(H[C]) a n3r3 n3 1/6 d /4,
− −
≤ ≤
FurtherapplicationsoftheContainerMethod 9
wherethenumberofcontainerssatisfies
log C n2rplog3n n2 1/6 d /5.
− −
| |≤ ≤
Hence,byLemma3.5,thenumberofverticesinacontainerisatmost
n
V(C) (1+n 1/6 d /6) m(r).
− −
| |≤ 2
(cid:18) (cid:19)
Thisimpliesthatthenumberofcolouringscontainedinacontainerisatmost
(n)
col(C)≤ V(n2C)! 2 ≤ (1+n−1/6−d /6)m(r) (n2)≤m(r)(n2)en2−1/6−d/7.
(cid:16) (cid:17)
(cid:0) (cid:1)
Thatis,thetotalnumberX ofcolouringsisatmost
X m(r)(n2)en2−1/6−d/7en2−1/6−d/5 m(r)(n2)en2−1/6−d/8.
≤ ≤
Now consider colouringsin the continuoussetting. Cut up each edge of the cube
intorpieces.Wegetby(2)that
(vol(Mn))1/(n2)≤ (m(r)+1)r((n2n2))en2−1/6−d/8!1/(n2)
(n) 1/(n2)
(n) 4 2 1 1 1
≤ 2− 2 (cid:18)1+r(cid:19) ! (cid:18)1+n1/6+d /9(cid:19)≤ 2+n61−d ,
asrequired.
⊓⊔
Remark.Kozma,Meyerovitch,Morris,PeledandSamotij[16]usingastonger
supersaturation result and a somewhat different container type of theorem, in-
dependently, parallel to our work, improved the error term in Theorem 3.7 to
(log2n)n 1/2,wherethe 1/2isbestpossibleasitwaspointedoutin[15].
−
−
It would be interesting to extend the above ideas to generalised metric spaces
(note that there are several different definitions of these), hence we propose the
followingpurposelyvaguequestion:
Problem3.8 Itisanaturalquestiontoask:isthereaninterestingextension
of the problem discussed in this section, when instead of requiring metric
triangles,onewantsmetricd-dimensionalsimplicesforafixedd?
10 Jo´zsefBaloghandAdamZs.Wagner
4 The largestC -free subgraph ofarandom graph
4
NowweturnourattentiontoprovingTheorem1.1.Asbefore,foragraphGwrite
G2 forthe“propersquare”ofG,i.e.,wherexyisanedgeifandonlyifthereisaz
suchthatxzandzyareedgesinG.
To simplify the technicality of the proof, we present here only the case when
p=1/2;forlarger p we can use a properlychosensmaller c and the argumentis
thesame.
AssumethatH isthelargeC -freegraphthatweaimtofindinG(n,1/2).Anat-
4
uralapproachwouldbetodowhatFu¨redi[12]did,workingonaRamseytypeprob-
lem.UsingtheKleitman–Winstonmethod,hegaveanupperboundonthenumber
ofC -freegraphswithmedges.If pissufficientlysmall,thentheexpectednumber
4
of copies of such graphs in G(n,p) is o(1), proving the desired result. His upper
boundwas, assumingthat m>2n4/3(log2n), that the numberof such graphsis at
most(4n3/m2)m,hencetheexpectednumberofthemis(4pn3/m2)m=o(1),aslong
asm>(1/2 a)n3/2and p<1/16 a,wherea>0issomesmallconstant.
− −
Theideaofourproofisthattheunionboundintheaboveargumentistoowaste-
ful;hence,insteadofconsideringeachH separately,wewanttoshowthatforevery
C -freesubgraphHwithmanyedges,thereexistssomecertificateinthegraph.Each
4
of these certificates will say that in a certain part of the graphone needs to select
(1 o(1))d n3/2 edges of G(n,p) out of possible (1+o(1))d n3/2 pairs. Since the
−
numberof certificatesis muchsmaller than 1/(probabilitythatthe aboveunlikely
eventholds),whichisoforder2cn3/2,wecannowshowusingtheunionboundthat
w.h.p.thereisnosuchsubgraph.
Soourfirsttaskistobuildsuchacertificate.FixaC -freegraphH withatleast
4
m>(1/2 c)n3/2 edges, where for the p=1/2 case we can set c=10 5. First
−
fixalinear−orderp ofthevertexsetofH,whichwewilluseasatiebreakeramong
vertices.Wewillalsoneedasecondlinearordering,asfollows:
Definition4.1 GivenagraphG,amin-degreeorderingisanordering v ,...,v
1 n
{ }
ofV(G)suchthatforeachi [n],thevertexv isofminimumdegreeinG[v,...,v ].
i i n
∈
Whentherearemultiplesuchvertices,thenweletv bethefirstamongtheminthe
i
orderingp .
Fix a min-degree ordering of H and let Y := v ,...,v and X =V(H) Y,
1 s
{ } −
where we will choose s such that the minimum degree of H[X] is large. Now
we fix F H[X] with the following properties. The first is that F is sparse, i.e.,
e(F) n3⊂/2/log2n.ThesecondisthattheindependentsetsinF2 approximatethe
≤
independentsets in H2[X] rather well. In particular, we choose F such that every
largeindependentsetofH2[X]isina‘container’determinedbyF,wherethenum-
ber of containers is at most r:=2n2/5log20n. The key observation is that for every
v Y itsneighborhoodinX spansanindependentsetinH2[X],asH isC -free.
j 4
∈The certificate for H will be the vector [Y,F, d s , r s ], where d :=
{ j}j=1 { j}j=1 j
N(v ) X , and r r is the index of the container containing N(v ) X, and
j j j
| ∩ | ≤ ∩
s= Y .Thenumberofcertificatesisatmost2n n2/2 nn rn 22n3/2/logn.
| | · n3/2/log2n · · ≤
(cid:0) (cid:1)