Lu zak ( P o − z nańa d A l − ta n − t a , G period − a)an
Vo−j tě c
h R öl − d(Atana−comma Ga .
a − l number k and v − e erypo i tvi e consta n
{0comma − one, .., n − 1} withat eat αn e e − l m e − nts o − c nt
o − n provd d n i l su fficien y − l l a
r neral k i t wa se l e d nt h e affi m ai ev y h e o ts a
t
n n g t heo r m o f Se me r´i [ S z 7 . A f wy e a saf er S zem
er´ d s aper w a p u l s h d , a en i d f f e n pr o f o ft h i re
u lt,base on re g d c theo w asg ven b y F urst en brg[ Fu 7 7 Si nc ehte n
, th e ma no p np robl mc o
c r i gthe orig in al q sti n o f E r os n d T uá nh a sb n t o n
dbette
l wer bo u n sf oth e si ze o f A th t guaa nt e th ex se n ceo
f a r hmet
p g r ssios o f l ng t h k n A. U nf or u nate y, nt muc
h a b en
h
com
a
pi s df or k ≥ 4 T hee x p ict e t m aes t ha tf lo w f om Sz e mer´ d
is o r
in ap ro fa rv e ry po o an d F u sen b r g0 s p pr ach d o esn op ro v esu c
bou n s at a ll T h cas e k = 3i sm uc h beetr und rsto d. Ro t ’ orig
g
nal agu me ti m pl s t a t t i sen ought oas s um eth at| A ≥ /nlolo
dt e
an h
be t l ow e b u nd t dt eh as b ee ng ve n ide p ede nlyb y
He t
e
B r ow n an d S e red i (se
H −B7]), w ho h owed t ht forco m e ab
s lout
c o nta n c > 0 e v e y su b eo f [n] w ih a tleastn /(lo n) ele m en s
co
tai n a n a ri met cp r g e o n o fl n h th r e provide d
is suff i cien t l
la ge
emee ’ st oh e 1m aihm 9 t cpro rss on , c mbna ori
Ke ywo dsadphr s Ssu s : Sz
num b e − r
h
r
r gu a − l ri y l − e mma ra ndo
msee s f − o i teger . i
g
es rch othe f is − t author p r − a al y s − u ppo t d − e by
FAPESP
(r − oc.93/603 – 1 ) an
Tomasz
byC
bracketlef t − one33]
e
t n ao a 3−term a r t h
oo i n 3t r
rt metc p r g es sion s C learly natr al andi dt f o suc
e
a se t Ris a n M − l ement s e t RM u ni f rm y selcte d ro m a
l h e M − e e m en
s ubse t o[n , wh e re 1 ≤ M = M (n)≤ n its ob e cho s ns uia t b y. Oru m
a re s ulthere c o n m s tih sa p p e l ng,i n iiv eie a.
F o r integes ≤ M ≤ n, et R n, M ) be t h ep rob abil−i tys p c − aeo fa
l th M − element u b se t − sof [n equ ipp edw i hth eun f r m m e sur e
. I nt hes equ e gi v n 0< α ≤ 1 and a e t R⊆ [n] ,let us w rit e R → α3 if
any A ⊆R wi t | A| ≥ α | R | c o tain s 3hyphen − tmr ar i h m e c pr gr s
o
i − son.O ur main res l ma t h en be s ate dasf o l − lw
s.
T
heo re m 1.F o − rev e r y c ns a − tn t 0 < α ≤ 1, the re
xists a
con s t − an
√
C = C(α) such t ha tif C
n≤ M = M (n) ≤ nt − h en the pr bab l t − i y
th a
RM ∈ Rn, M sat s − f i es
R → α3 te ds to 1 s n → ∞. o
i
e ma
( n
Fo m T h − e re m 1,t
om [n − bracketright ind p e den lyw t − i hpro bab i y . hu , we
w r − imte R n p) forth p ro ab i − l i yspa ceof t s ch R, e − h fo ragi
vn s t fR ⊆ [] th e p ro b bi l − i t th t Rp = R i s p | R | (1 − p)minus − bar| .
Th orem
two − period F o r
ev eryc on tant 0 < α ≤ 1, h − t e r − e
e i − x sts a co n tan C = C(α su c ht
at if
C/n ≤ p = (n ) ≤ 1
th n thep r bab l − i ty th
Rp ∈ R(n, M ) sa t − i sfie s Rp → α 3 te nds t o 1 √as n → ∞.
No et that T h e o r m 1 a n d e e
i − thmei−tcp o g e − ssi nsin R M ∈ R(, M
i w t h lag e
rou bab l y , sm a l − l e t ah 2ε2 | RM |, n dh en c − e a l − l o t e m
mayb de e roye dby deletint g a mos 22RM | l − e m ents fro m RM; n other
wo d , i − w t a − rg
√
onR M → α 3doesno ho d f r α = 1 − 2ε. C lear y, ais m ar p eno mn
o h p n o r pR w i h p = pn ) = ε/ n
O urr sul s ab ove mi md t e − l y m py th ee x i tence o f “s p a r se00 sts S =
Sα s − u c h th at S → α3 fo r a n y f ied 0 < α ≤1. Th e f lo wi n g
res l t ma e − k
th ss s e rto n pr cis.
Co ol a ry 3 . Sup os e tha t s = sn ) = o(n1/8) n d g = g(n) = ( log n as
n → r ∞lT en , f o e ery
fix ed alpha − greater
a − t o − f r ever y
n ≥ rN thre exist s S ⊆ n] s atisfy i − n g
Ssa
n→α3 f oa r
wh i h th
tf lowi ngth e on d ion sho d .
o
√
k +, ... , k + lm} w i h
k ≥ 0, l ≥ o g
(i i )If F = F (S i th e 3 u i
o rm h y er r p
evrte
se
S wh
s
o hy p e redg s aret he 3t m
a r ih m eic
pr oresio ns o tai ned i
S, t he n
h an ocy cle o fle n
g
I n words co n dt i n s( i ) a n (i − i − parenright a ov e s y h t th e et S i
te e − sctsa n
arithme t i − c prog esoni a smal u b − m
e
i ns no f our − hyphen mr − a i h met c p − r og r s − e s o − nperiod − s
r
C ondi−t on ( ii ism o e c m − o b i − n a o − t rialin na ure an d says hat
the 3 − e − r marith me icpr og e − r s io s onta n − e
in S ocall f − o r m a r e − e− l ike stru tu
om ewh at urp ris n g
i
re
Le tu s m ar kth at t efo ll own gex t e n ionof S em er´d0 s t h o emr e alte
do C orol ary 3w asp ove d i n[ R¨o ],t her eby s e t li n ga p obe m ra ise b y Sp e ncr[
p 7].Le t k,g ≥3 b e fix d i nseer and 0 < α ≤ 1 b e afi x
r e .T heo e m 43of [R¨o 90 a se rtst a th e nf oa n yl rge en ou h n e eis
o m
sak − u nf r
h
rth me c po g s sio s ,s c hth ta F co n tin sn o c ycleo el n gt h sm ale th a g
|
b ut ea h sub s e A ⊆ [n] w th |A ≥ n co nt in s a h p ee dgeo F. F o thr
pro le m s a d r s t i t hi dir e ti on,see G r h a a n Nˇ tˇ [ G
86 Neˇetˇ a n R öd l [ N R 8 7 ] a n d P r¨me an dV o igt[P
V 8 8.
e
N t et h C or l ary3 st r gthen st h b ve r e s l to f [ R¨9 0int he case k = 3.
Th ep ro oo f T he re m 1 i s u for u n@ ey q ute l n g .In te ne x
tse o n w e d sc rb e o r e n ra l a p p r a , res sni
g h em an ide a s
v ve
and ignorins ev e al q uit t c hni cal
p art
s W e hop eth tt h o uti n e of o m
o
ew
thd p r se n e th r il b e o so me u e i fol ow n g he cual poo T
e
h e or g n z at i onof th e pa p ers as o d c u ss
di n th e ne xtse t n .
1
. Outli e
f
th em
e
t h od of proof . h e m ain l mm
a i nhe proo
of he re m 1ns Lem √ ma 19 . Inesenc e w ha tT hi sem mae s y ss tqute
sim pe T. Asu m e C n ≤ M 0 = M 0(n) s ≤ n,f o so mela g e C > 0. D
ire a din
som et chn calii s , L e m a 1 9 st test heo lo wing : ifw eco ndiion or no u set
R 0 e ∈ Ri(n, M e ) sa isyin g acer tin “ spar sn ess con ito n , tt
i h a t RM 0 a 0st co n ai n a n rt h m cp gre s ionof en t h hreei
a tmo ste p {− cM } w h e w m ay m kec arb itrry la rg b y pic in
g C ap p − r oprt l a − l r g
Th eer m 1 iss ownto o l wf ro m Le m ma 19 in t w os eps S upp s −1 is l a
ec n a n t it h re sp etc t o a gv e nf i x ed α > 0 a n d M = M ( ) ≤ αM
W e a ima tso w i ng th at RM → α3wi tohpr b abi itya ppr o a h i ng1 Ou r f ir s
epc o ni s t s o f a q c k ca u to n ba e d n L e mm a 1 9to de duc e t h t
e i lR M ∈ R(n, M t
T h u s all u r f fo t g − o i op ro vi ng L e m m a1 9
An
im
p
t − ra n
ttooli
t he p r − oo f wil lb e a v e sio nf S z e mr´e di sre g ul ar t yl m m a [ S z 7].A s
wel kn o − wn, t his san imp ortant g aph th eor te i alc o mpon en tof Szem
e e − acute − rid
p roo o f h − is t eo r e m o na r t m − he t pr o e − s s io s. I tturn outth t − ait
ism o co nv
in g n id eao f Ru zs an d Sz m e ´e d i RSz8] se as oErdos,F 0 n k l an R
d[
ö
E F R 8 6o r Gra ham0 a dR¨dl [ G R 87)f or ev erysu se t R o [n w
u
c o nsruc a g r pa h G(n,R )t h a,r
g hlys pea k i ng , h@ h e p r operyt hat
o
s
,a s
co 0 t i n s a t rang e ( m r ep reciel y “ opntan o s ” tr an g e )if an d nl
ya
y R con t a ns a3−ter mar th m ec pr gr ess no ( moe p r cis
n “ r it m et
e
i
i
t p l ” ) Lem m a19is in f ct sat e d n te ms f s a rse gr aph an d sp on t
n eo usra ng l s , nd t s er s t h at spr e g p h sree of such tr a n g l e ar
ex trem e y rear .
U n o tnu a tly t hep rof of Le m m a 19icsquite o pml e, a n d w e
sh ll n o
a tte mtp t o g vean on− t ec nca lo u of t here P rob a bl a y s uc htske c
w o u l d ailob eo mu h h l p. N o netheles w e rem a rhkt at th e a rgumen
t−o
beo w is d ivid di n o e ver al t eps which are
al a g e ext n ti, pnde en d e o f
,
one an othe
w
T h e o rg an iz t on o fo u r p − aperi a s fo lo s I n S e c on 2 we i nt
s
ro d u then o to n
o r egu ait y − comma u n f or m i tya n ds parse e − nss o
f gr ap h , a n − d t − sat e ve r s − i n o fS em e r´is r gu la y − t le m m a fo
su taby s − p a rse g a − rp hs t oet h wth a f − ew rl a te dr sults . W e ta
t Se c o n 3w t h a na na log u eo f atherem o fR uz a and Sz e meréd i[ R
S z 7 8 f o s p a se r − gap hs (f . Th
emm a 9, na d the n g vea ni m p rt an b utra he tnech ic l em m a L e mm
“0
10 , c on ce n i g the ex is enc e of c er ai n t ructuresw e c all
5 o w r s00 in e lcoo
“
It
s
hat
u e d s pase g ap hs .
i in p ron gL e mm a 10 t
we shl m a k eus
o f S e m r´di0s r gul rit lm ma i nt hef o r mg i v n n S ecton 2
On e f t e mi n por bab l ti c ing e den t i n the p r o f of Lemm
a 19 i gvn n ec i o n4. Roug hl ys peakin g w e sh o w i n L m m a 11tha ra
no i duced su b r ap h of a b i p r ti e un f or m g rap hc on ta nsw
ith ve y larg poba b i tya f i r num be o fe g es. I n Scin 5 w e giv ea s i mp es
uf f i ien c ondi o n fo r ar e gul arb iprit eg r a − ph t ob eunio rm .I nSec i − to
ugr a p h th eertial rs ul ts of the p rve i ou ss e ci ns t o s u betof n] w
def inetdhe“ i e nceg rap h 00G R = GR nf o anygivn s u s etR o f [ nan
usn g r sult
o m S e to n 5,s ho wt ha t R i sa a n dm
s to f ut a y l
o
f
ar e xp e teds i et, h eni sdi f f e e nceg ap hG R is u io m w t lareg p r bba ilt F inal l
f
n
y t h s ta temen t an dp r oof
ma Le mm a 1 9 an d th
o ou rm ai l em
r
p hs an dS
ze
m er e ´di0 s l e m
m
(
a n d e(G) =| EG )fo r itssiz |E (G. F ur t e r m or
et U, W⊆ V
bea
W
,a dset
(U W
)=
eG( ,
W ) =| EG (U
th e pai r(U , W
d (,
)
i
= eG(U
GU
For 0 < ε ≤ 1, w e s y
pl ε eg lar , f o r a l U ⊆
| W 0| ≥ εW
h t
U
W
r ap for t h e orderof
).T
f ine d
d
,W
)/
sdG
(U W
)
y
(G) .
| U || W |
W
den
G
n2
th epai r (U W ) is (εG − g lar
or im
an d W 0 ⊆ W wi h | U 0 ≥| U
| a un d
we hau v e
f
l
is(l,)
W or
| dG
( U − commaW )U−dG( 0, )0|≤ ε
th e 2
p ri of d − i s inc vert x c s − ases oft h − el − p rti o − i
n o f Gar e ε re g − u lar W e s ay th t a p rt t on Π = (V i − parenright o
ft h e v rt e x s t V
= V G( )f − oG
i − s(ε, k
o−f
a
Π. We a yt hat
e pΠ ionalto
ar
n
of on−
p
b titio,eve yn Π n.
F or exc
suat
i
th
he p
he (,k )−
) 0 isof cot
sus ( Wo
of naΠ in0in hedVth ince swme
h1mem we
be lca as= bp f ar t i,to n
a
i
Πi0 ro
h er ie be
a n e u i tlabe pa t i io n a s we w e re u r y o 0− e cp
io n a cl so f Π to be c nta n d in s o m e ik xc e t ion l cl s s o f
W
Π.
esa y t ha t an (k)
k
eq uia bel
p () i o n Π = i(V 0 i ε, ; G ) r e l a r o r sm py (, G r g l
if a tm ost ε2 p ars ( V, V j with 1 ≤ < j ≤ k ar e no( εG− eg uar
l
or giv e − nb >2 a d 0 < η ≤ 1, w es ay t ha t G i s
Fi na ly,
b − parenlef t,η) sp rs
o n
W
i , f ore ver disj i
t ai of s t U, W ⊆ V su c t h a t | U |,|
| ≥ ηn, w
e h dGU , W) ≤ b T hu , r ugh y sp eakn g a gr a p hs parenlef t−bη )
s p a rse fa l l f it
f
,he
d
n
0 d en
la rogn wsta tuc
saor eSz noe emm
mmrt af o r (b) η −rs prsh e grel f a phW s .
e o e ursext n hesp
e r´i cds l
i−o
L
e mm
a 4 . For a y g v n
ε > 0b > 2, k ≥ 1 an d s ≥ 1, t here a r
c
ons a
n t−s η = η(εb, k0, s) > 0a n K0 = K (ε, b zero−k, s) ≥ k0t
h a − t d epen o nl−y n − o ,ε , 0, and s for w ichth efo l wingh o l s . F o r every
(b, η)hyphen − s p ars gra ph
G an
everypar it i − o n
(W )0 o f h − t ev erex
s e − t of
G, t here
x − ei−s s − t a (epsilon − commaGhyphen − parenright
r e ula r (ε, k)− eq it blesubpart i ionof (W j)s−zero w i t h
k0 ≤ k ≤ K0
c
lo n t e am e i e − s a t
g
c
e − h pr oofof Sze m eré dis−quoteright o
n al r − eult [ z 8 , an he
ew
r
n
eo m t i t h re . A s am a t − tero
Thep
oo
of
e
m ma
4go
s
t hatG
m di tely folo s t h
S
s eq ue nc eof gr ph sd ef iedo n he am ever exs e t V,a d met G = i = 1G
b etheir u n i − on , i . e comma − period th g rap h on V w − i − t h t he
edge set i = 1 E( Gparenright−i
We sa
.
p
1..,G m i s (b η − )s ik r eif Gs (b,η parenright − hyphen p ar e. u − F t − h e r m o r , n − a(ε
quta epart o Π = (V )0 i s( , k;G1, ..,G m− egu r if it i (ε ,k Gi) re ul a f
o a 1 ≤ i≤ m
One c a ne ai−ly de duc e fr mLe mma 4 t h@ ev e y( bη)sp a s − res q u−e n eo
a
g ap hs G1,... , Gm ad mit
n (k G1 , .., Gm)− eg u a − rp a t − rii−o n, r − po vi d
s
η i sm a l l e no gh wi t h r espec t t o , ε b a n d 1/k R u g hy speaki
ng w e f irs
ch oo sea r a p i d lyd e − cer sn−i g s eq u en ce ε= εm ≥ε m − 1 ≥.. . ≥
ε1 > 0o fc on s tants ,an dt h n ni v e
f ine r a nd f in r part tion of V. T bep rec s ,Π i i sre q ir edt ob e
s ubp ri o iminus − one for al 1 < i≤ a d
(1
r q
Πi ≤ i ≤ m ) s e ui r
to b an ( ε, Gi −e g lar p a r it o o V
w th a“s mal n u mb e
uar
oc lass (ass m al a sLe m ma 4 c a n g
ne)C oi th i 1≤ i≤ m
ca r ful ly en o g h,t ef i na pario n Πmof o ur eq uen e Π1,..,Π mw b
u
(ε kG 1, ., Gm)reg
larp a r t nw se ek
It w ill b e mpo r tan la e t ha t − comma r n y 1< i ≤ m , h
e pa ti toin Π
a bo em a y be de tem i ed s oel
l f r m Πi − a n d Gi. Ino h e
w r d s th
gr p h s Gi + one − comma..,G m pay no rol ei nt e − hde f in t i ono f Π i.
W enow ma k − e th e abo vei f − no ma ldi cussi−o np e − rc i s e
. T − hu , let k an m ≥ 3 b e n a − tr a ln u mb rcomma−s let ε e
asequ
at 0 < ε1 ≤ . . ≤ εm < 1 an d etG 1, . . G m b−e a phs with
t he m − a vert xs e V, w here| V
|≥ k0.In t he e − f ini t on b o − lw w e
shal ass m e tha th ee toaf l l e q u ia lep a ti io ns of Vh av e b e en i v e − n a
.
˜ 1,. ., widem
˜
Th e (, 0)ca n on a l
eq ue nceo f a riti no s wide
G1 .. , Gm
fo
i s e f in dr ecur s vel inth e f l ow n w ay
( ) A m no g a l t e (1kG one − parenright reg l − uar p a r t − ii−t n s
e b t h efirst o n e ca co r d n − i gto ≺ . 0 ( ii A s
whichm inim ze k ≥ k let Π1
˜ − i1 ha salre ady bee
u metha t 2 ≤ i ≤ m a nd h atth epart iion wide
˜
d fine d . T he n w ee t Πwide
bethe ≺ − rs t (ε, k; G..., G eg ula
su b p a r to
˜
ewide
l
fi
i
−
N
th ta n ypa ritio no f he v
s
ε, 0)− c n no i la e uen ceof par ttor enasdo esex sta n 1ds
ot
bg uo v , an
of o urs
unique by d fin i − tio period − n
s
i
i
c
v − er y ε > 0, b > 2, k0 ≥ 1 and
m≥ an
m, bε, 0) = (εcomma − period..εm ) w t − i h 0 < ε1 ≤ .. ≤ εm = ε s ch
th a − tη, K
a nd ε de e nd
on y
o n ε, b, k0 an d
m a nd the
f lowin gh
lds
.
For ev e − r (, η)− spars sequenc
ofg aphs G
˜
parii n − s Π1, .., Πwidem
a so i a d
i − wt G1, ..Gmon yco n tan spar tit on
f
o i z s b o − un e d
y K 0 + .I nf t − c, w
eh av e
k
e ≤ .. ≤
1 ≤ | Π1
t at t
e m
|
Πm |
≤ K0
+.
ma e e nd o ly o n m , b, ε, a nd k. I
h e
fa ct t hrou g p p ew e s h l a s a or an y g iv enm , bε, an
k0 a si n Le m m a5 w h a a f i xe d
vc or ε
o d
e h s es ti−o n w t h wt oi−s m p l − e ob r − eva
ε−re g u − l a t y d al wi h th d strb tion f − o ed ge b e e n “
a g ” et N o n e − thele s − s,it t un so ut ha t e ac h ε− uif o r − m par ie
t
g raph G c t n s l rg e3 εuior m pa rite u b gr a h Gsu c
haeah
v−e
h
e
e
a
r x of Gha saf rl larg
d e r . I n fc t , m o e i s u a sh o wn
Fa ct
6 . Supp ose
l ≥ 2 a nd 0 < ε < 1(5). Let
G b ean
εhyphen − u ni for
(l + 1)− partit gra p wih (l + 1)− p ar t − i t ion
V (G) = V 0 ∪ ... ∪ V l.
T
he
th er e x − i st
subset s
Vi ⊆ i (1 ≤ i ≤ parenright − l such
th a − t for eve r y 1 ≤ i ≤ l, we
h v | V i ≥ (1 − l − epsilon)Vi | a nd f
oreveryv r − e ex v ∈ V i weh ave
very
j 6= i( 1 ≤ j ≤ ) In
a
i
u
r
t he g r
ph
G
i
du e d
in G b
or uniontext − l l. p
rt
c
a , a
n
c
i = 1V i i s
ε− i
r
m.
P r o o . S i nc e pro fso fvery simila s a te m e − n t scan
no n t h e mi e id a b e h n dh a r g um e
th ec on it o s ≤ w se ek . T h e no n − e c an eas i y s ow
th vt − a b e − cau s o t e ε u n i o − rm it yo f G, thsp
F n a l − ly , sinc eevery gr ap h G on n v er i − tc − e s co n a − t i n − s abipa tr i esub rap
H wh se v r − t ex l − c a ses a r − e of card n a i − lty bn/c and dn/ 2e
and s u c h t − h a e(H) ≥ (G )2/ t ef o − l lowin gfa t − c is ani mme i − a e
c nseq uen ceof the defi ni iono f a (bη) spa r − se grap .
| V (G parenright − bar + 2 v rtces conta s − n at m ost b | V (H) | 2e(G) /V (G) |2 e ges .
No
te
0+
he ε i n t h l
The
[ v id for( b,η)
h Rus − −Semer´ditho re m tha w h a n
e st at d a f olow s
ore m 8 . Forev
ry c − o n tan c > t e − h r e
su hth a
tever
3 decom posab e g
ap
sco tain
a
prs
eed (and
gn
ra z e b elo w m yb
b
x
s − t s ac n − ost−a t − n δparenright
−c
s−t
h G with
tlea c n2 e
ge
leas−tbδ()n 3tra − ingles−period
Wew
u
ldk et
o ppl yas mlar es ltfor grap s wh charen o t o d ens
Un o t unatel , i nthis c
t
dcompo
s
−
ab
gra
phs
G
with
n v t − r c s and at le s t n 2 exp
e
√
(−3
on
on a−tinonly e
se e , e . g , T e
edg87] ).W
esw h − icheacrethus
f −o (G)/3 r − t a g − l es
r − oem 6.6 in GR
um eth od wi lc o nss t i n pr ovi nga p r o a b lst c vre s on o T he ore m L e
mma 19 ,a ss r tng t ha , in s om e s ense “ ou ne e x amp ls ” a ab o e a
t ow o r f or aw hi eb e o r we m ay st a t e a
r are.Howev er, we nee
1
d − p − r o v Le m m a 9
We sta t wit h ar e ul s ai ngt h ati a 3 de c mpos ble gpra h G ad m
a ε− r eg u r pa rtton t he na hlt o g G m a y c ont ain onl a fw tria l
itmu tc ntain m a y“d ns ” ti lesof artitonc las e. T o em p ai
eth di eene be we nt r ng ls o the o n e ha nd an d ti ps o f p ar ii o class oth o t
t
Th us, l e G b a (b, η)−sp a s e g a p h o nn v rtcesa ndl
=V( i) b
e Π
e
i a
e
e
e an ( ε k − ru tsb (εG ) r egua r p tt o of th v er te s eto G. W s
k
tat a pa i (V, V ) (1 ≤ r < s ≤k ) i thic
if t is (ε, G− eg uar na d
eG(V r, V s) ≥ V r | · | V | (G)/(50n2.
W e s y t h a ta rad (V r, V sV t)(1 ≤ r < s <
h
rs (V r,V s(V rV t ) a n d (V sV t) a et i k .t s h i k
lt r e , r c
.
Le
mm a9 F o r ev e y b > 2 t er e e
xt
co ns a n t δ = δ() >
3 d ece m po a eb−l gr a p h G, h i f a Π t ise an (ε
re ula rpar i ion o f G s − u c ht h a t zero − two0b ≤ 1, k ≥ k, an 0 < η ≤
m in {epsilon−comma 1/ (2k) }the n Π c n − ota−i ns at
eas
δk3t
Proo .L t k = k0b = 4 0b n d
δ = δ(b) =b( .3 b) > zero − comma w ere
i a sg ive n y The orem 8 W e s h − al − l show tha tth e − s e vau s − e
wild ofo ou
le m m a . Supp o e 200εb ≤ 1 a n dlet Πb ea n (εk)equ i t a ble (ε, G − re g − u
l partt on of a ( , ηhyphen − s arse 3d − hyphen e ompos bl g a
Gh , w ere k ≥ 0 na
0 < η ≤ m i n {ε , 1/(k − parenright. The nt he follo aing asser io nsh
>
εn2be(G)n2 ≤ εb − e(G ) ≤ eG)/20
i i ) t h enum n o − tε− rgular s e sstha
k
( n
n
2b − e
2
(i v )th e n m b r o f ed g s
a e n t th ickis le ssth ( n )
2
be we n
nk f ive − zeron2
G)
k
pais(
n
V
rV s )(
1 ≤ r < s ≤ k) t
h
≤e (G/10 0.
e ns c e edg sfr t0m he e Gdge et eof th besp e a ni
g st u
gr
pa h
0
o Gf .s
Let F
t e o t
a
0
t−r−h
t − tG − e
a − c o siton
n−tain aΠ l e hs t a tl e
,
p
e
(V s1 ) , V ( t( )). L t u s1el t e o m G a le g es b w ee he et
o
s r1 V and V an d let G b e t he g raph
b ai ne2in t hs w a y i c 2
r
a
n m b e of e d g esw e de let e si2 s m l e r h an 3bn/ keG)2 ≤ 3be( )G /k
we 1 d e st o y a mo s t be(G )/ k < .3e(G)tria n g e s f r o m F. T h u , h eg ra p
i
G 1co n in a t ran
gle a nd he nc e he a tit o n Π, v wed s a pra t t
on o
G, ontai sa t l e ato ne 0hick tria W e r pat te proc du e ab oe a n
o b t a in a eq u e e G = G ⊃. . ⊃ G o l p an nn g su g rpa s o
G wi hG su cht h a Π,v
ew d as a pa rtt on o fG, c n t ai n s no t h
c
S
c
i
t i ad . i n e2 ev er ysep w d e c 2as et h e n m b r of t i naglse i n F b a
tmos 3be G)/k w 1 ha v elk ≥ 01. k rb. H s n e, th e gr p h G (Π w ho sev
e t V ..., V r n dsV , V (1 ≤ r < s ≤ )a re j oin 2db y an e
e ti e ar e the
.
dgei a nd o nl y th ep a r (V , V ) t hk , a na t l e ast0 1 k/bed g edi j o n tr i a g e T
h s , b Th eo re m 3 a d o u r c hoc e o f δ = δ(b) = (03/bparenright − commath
e gr aph 3G(Π
co n a ins al e − a st δk t i − ra nge . C o s e quen tly th e − ree i t atl e st δkthic
tr adsi n Π a n d Lem ma9 fo low .
W e no wtu n t th e m a n l m m a o fth s sec t − in − o, Le m ma 10 . As
alre ad m ntio ne−d i nS e c t on 1 , t h s − i isa r th ertec h n i ca lr su l − t, an d
bef r e we m a s − t@e it wenee d o n − i tr duc ea few d e ni t − ions ,i n cu−l
ding the def ini−t onof
“ “ 0 15 o we ” ina tne dge −c − o lour d r − g aph . Le tu s say that
as equen G − cone−e ..., Gm
ofgr ap s ont esame v rte xset i − s
tw
nk o f a b aa nc e s eq
e = (i − G) imbe= a ba a c e − d se q eu c e f − ogr a − ph
G
˜ 1 , ., wide
˜ m is
| V (G). S pu p ose a so ha t wide
a r t i ion s a s co a et dw th G1, .., G m fo rso m eg i v − en k0 a d − n som e
ε1 ...εm ). Th en , fo r a y
con ta n 0 < δ ≤ , a (δ, k0ε; G)f l
o er o r , for
ho t , a (δ, G − tildewide − parenrightf lo w er , o ns s s − t of hreei nd−i ce s
1 ≤ w(1) < w2) < w3) ≤ m oge h wr it−h a v t − r e x v of G a n afamily
{(X (i , Y( )) :≤ i ≤}g fp is
)
f no n −ex−c e p
inalele
men s
)
X an
d
Πw()s
Y io f
uc h hat
(i)2g | X() |= 2g | Y (|≥ δn,
( i i ) t he 2g sets X (i − parenright, Y () (1 ≤ i ≤ g a e a l d s ti nc t , ( ii i a l lp
ai rs (X(i − parenright, Y (i)) a r e (εw(i, Gw r(1)− e gu a r a n d
)
G
e wparenlef t − one)( X, Y
≥ δX
(i
)· | Y
(i)
(i
parenlef t − i)r
| parenlef t − eGw ())/(
l
0
62
n
=
iv ) he v − er ex v isjo ned t o a − e c h Xi (1 ≤ i ≤ g) by
w(2)/(106 n2 = δ | X() | eGparenright − slash(106 m two − n) e d g esof Gw(),
6t2x v is j( i − ne d o e ah−c Y 2)( 1 ≤ i ≤ g) b y @l e a
m
d
(10
n )e g e so f G w)3
1
Le
mm
a 0 . Let b > 2 be g ve .
Th na h
e e e i ts i n ge s
m = m() 3
a nd k = k 1 b, a d a e
l n um b e r 0 < δ t = δr(b) ≤ 1 that d
th t , f ra ny
0 < rεa≤1, h e eexi s
a
c o sn a tn 0 < η = p
η(b, ε) ≤ 1o f o r w hch t h e f o lo w
n h o l s
L e t ε = r ε(b, εk0). I
˜ = (Gi
Gwide
d − ge o − c lo rng of
a (b − commaη) spars e hyphen − three d e − c o m po
parenlef t − epsilon, k)c − hyphen n o
ica s e − qe − u nc
ofpa
i t−i
˜ Π1, .., Πm aso−s c a − t d i − w t h G = Gi)m − equal1 da mi ts a
on s wide
˜
(δ, 0ε; Gwidehyphen
− parenright flower
R i e m a rk . In h e e uq l , w hen con s e in g b, η) p a es s e q u e
cn s o f
˜ G as above , we s s h llo t ne s ayt h a t “ a (δ, , ε; Gwide
˜ − rf low r
a phs wide
x st es00 or
at “ ) ] T J / F 13suppress9.6suppress T suppress27.792.52suppress T D [ ( e
0 ) ee i
r y
t
Ghc n t in s a (δG )−f l o w er.Ins uh c as es,we ae t ssum ni g t
a
ε
istho
biwide,
˜ lw
o dw
r
h er e ε = eε m,, , ε, a k
w i h m, b, ε a nd k . r h oo f o Le m
b e as g − iv en by Le m m a 9 . Se
f .
m
= m(b)
=
ceilinglef t − three
m
a
10 L
·1 06/ δ(parenright − be,
δ
et
δb) >
= δ(b) = δ
0
b)(160m3)
=
> 0and
k−
) /
0b ,δb) (80 0 m )} N w l e t us i o o
l e m m We ma y a ndslh lass u m e h t η≤ nm{ ε 1/ (2K 0}O uraim
s sh o w th a t t ass r to n hol dsf o rη = η ( b, ) = η(bε ,k 0m) gi e a bo v
S
−
T e r − ef or e, l t G = (Gii = b
m l n d − e me edg colou rin
go a (bη)− spr 3− d e o−mpo s a b e−l graph G = iequal−one Gi. Wen e e to ver
fy thatu d er t e−hs c o d i t−io ns ,a delta−parenlef t, 0 epsilon − semicolon G)f l
ei nd ee de x − i − s t
˜ 1 comma − period..wideΠm
˜
Le wide
be t he epsilon − parenleft, k)− ca o n −
ic l − asequ e n ceof pa r − tt−ii n s a s o c ate wit h G, ..., G m W e r
n
s − tc
e n tr a t oe u r a tt ent on o n t e g a − r p
o
1
˜
G = i = 1Gian dt h e p a − rt t on wideΠm.
e−W s−h w−o t h a−t
ea h o at le ast ha
o − f thethick t ads (rV V t )n − i m, wh o e e xste n e i − s g uaran eed b
V
Le
mm
a9
h a s t − he
p r ope rtyth t − a
we
may a sign t
s
V, V ), ( V , V) ad parenlef t − V , V ), so m eth re e ds tint c l o rs s o that
.
≤ m, t enitco nta in as u sta n tia
ifa isd f om
is a wsGgn dc o l o u r 1≤
f eg
nu ber od
L e tu s em
tef ir s − t t e − h t irasfo w hich a n assgn me nta r s
i i na
wss e ki ntp ss ble Gi v n 1 ≤ w ≤ m, w es a y tha t a t hi
k t ria d
(V, V , V )t o
˜
ths ε − re g u−l ar pa r it o − i n Πwide mo f G s d om in a − te d b y wi,rpu
t s ng ur = tV r |= | V |= | V |, a te a t w o t ofth e t hre e pir s of
se ts( V , V , ( V, V )an
(V V
u2e(G)/(150n2) ≥ (/(2km parenright − parenright2e(G)/(502) =
)
e(G)/60k2)
n
(
0m
˜
w, w h e r e km= wideΠm
− 1 ≤ K0 .I f ah c k t iadis
e
a
n t do m − in a
by an y w (1 ≤ w ≤m) w e s a yth @it sb la n c ed .S ince
t
ev er yver ex f G has de g ree 2e (Gw/n = 2eG) /
at e da b y i w isle s ts han f Πm . C ne q u e nl ,t u m b er o f h i
ed e so f
G
2
kme(G/(6002m)
≤
1.5 · 06km 3
m
2
≤
δ(bk3m
2
m,
) km
we
od
s−
s o t h e nu 3berof t h c − k r i s − dd
m)si3 ls tha n δ( )km / T hu s by L
ph e − r
(V r , V s)of p
num
berof ed
a−g
o − minate
i − to cbe
on
d
b ys
ome w
an
˜
lasss−eo ΠV rwidem
(1≤
a
w≤
wV
| t |· V | e(G w)(15 0) = |V r||V s|e( G)( 50m ) . W e y t a t V ( , V s
t
V ) is (w(one−parenright,w (2)w three−parenlef t) −coor bl e ,w her
wi)i( ∈
e e
r−e
{, 23) ar et h
d istin c − t c ol u − ors f r − om
m h an ys u h t i d i s
(w(1 )w(2, w()(w −mt−parenrightbig −ih 1 ≤ w( 1) < w( 2) < w(3) ≤ m s u
˜
htha twide Π mc onti a tle st (b)k m23 thi c t a dsth t ar ( w(1 ), w 2)
ol ou a b l − e.
Th u − s, the eei−xs a on −e x − cep i − t o a − lp a i − tt − i oncla
˜
s V o Πwide
n pa r (U m, W i1m)( ≤ i ≤ f = dδb)km/(2m3)e) of no n m
e xc e − pi − t o n al part i − t io
e
c a s − s es f − oΠm
s u h h − t at , f − o r al−l ≤1 i ≤ f, t − he tripl e
(V, U m, W i − m ) s ti
fie s t h − e f ol o wi n g co n i o − it n − s :
( i ) the pir (U i, W mi) is (ε, Gw(1)− r gu l a r , a n d
m
G
w
)1
m
W
,
m
|
)
· W
≥ Um | |
m|
parenlef t − i) t e − hp a − i r (V, Um i) is (ε, Gw
e
a
wG (V , Um )
W
V,
)i i, G
ewG )( V,
|
i
· m
|
m
i
) i
(
)
−
2
(G(1)/(150n),
(2 − g − e
w3) −e gi u l a,
Wm ≥
i
w3)
u − la ,
ad
2))
d
w(
h wv−e
6
r t c es o f V
a r e t j o ined vb
e
U im| e2 − 1())/ 300 n) e dg s o Gw (2) t oU m n d
e r − e i s Gavr − e3t−e x v W n V T mand aset Ξ = {(m Um
w(1) − r h pars (U mW m o f c a − r d na t
|Λ | ≥ f 22 ≥ δb)k m (4 m
ui h th a t v is j i − n ed b y t − a eas t
| U m | e − parenlef t w − G 2)) (3 00 n e dg so iG wparenlef t−two ) t
s
(U mUm iani ,W m )by∈Ξ atlea. t | W m | eGw(3) ) (3 zero−zeron 2) e ge s o f Gw3)t
˜ s e − n ce o s uc h va re
o W m f or a We sh a ll s ow t hat the e iwide
e x v a d−n u c a t Ξ
i p li es t h − e x it ne−c a ( , Gf l o w r . T h e t h e coo
s a s so ci t ed wi t
a
hb
d
comma−parenright
low
rew ( 1comma−parenright
it
rittw(2i−o
f afthe
weave−r el−a f lowe r we ew oi snlh lyco e ns−i−d
t
eredt hep ap a
est
˜
h aml−f f − oF t
˜ wide ( 3parenright−periodO urer ons − ayp the eob emi ot ont o h ts th sh altry tor
n Πwide
m
,.wh
n
de n−i
l − a te hep ro pe r i − t es of Πm w i ht h os o f Πwparenlef t − one). I n rde tr
do th s − comma w es ha l − l co si e th e graph
˜ m Ξ) w hose ve tex set s − i he s
Gbparenlef t − P iwide
of non - e ce tiona lp arti ion c − l a s s e sof Πm, w i htw o su h vert c − i
se U m, W m
e m, Ξ)fa ndon ly f parenlef t − U m, W m) ∈ Ξ .
b in c − o n ecte dby a nedg ein Gb(Π
b Πm
e
N ote f ir − s t t h t − a G(
, Ξ) has N = km v e tces an dat lea t
cN 2 e dge
w here
e
Πw(1
e m isa
c = δ(b)/(8m3). Furtherm o r − e, i − sn e Π
su parti t on of
˜
on alcla s s V m0 of Πwidem,
not a l par bw(
n
see that a − l lof them con taina t mo t km /k(1) elem e tn s a d − n
alttleargu ment sho w s t − h at at e − l a s − t( 1 − 3parenright − epsilonk w )
o fthem h ea t as km/(2kw(1
s
œ l
e−l emen t . N thaow su tthepf olowi−n
≥1hol d 00/c colon − snd
.T whtnh N i
g
f or any ε ≤gra cGb 0
/
m
o
oa ssh
tle as
pleer cace
c uaan−d i
cN 2 e g e , a nd e ery p
ar i i o
nto
at
bΠ =( V )iequal−i 1 o f the ver te x s t of Gbi
c la s
p a r t − io n
cl s e ssuc hth a t a l th e
mos−t 2 N
1hil
esV
h e − v ca r i nalt y
and not
N/2l)
the e e − x
st
a t le s c/100 pa r − is{V ,V b0}o f
|
h@
ses|·|V 0f | Πbs10e uht
cdges . Hence V,|, si 0|c−e
N/ (2
≥v−e ry gr)a np dho V
dis in ct
p r tii
joi ed b
est kve r−ti yaestle
and aa
clc−barVb a
n Va 0mo ar
le a s − t ck2 e d e − g s o n ain sa ma tch n gof s ea tlea t ck /2, th e ab vepa
b must cont i n a lea s t g = ceilinglef t − c/200 d s − i joint
r i − t i o Πbo − f G
bi, Yb i)( 1 ≤ i ≤ g
pa rs (X
s ch t h at |bXibar−comma Ybi |≥ N/(2l) an d suc ht ha the n u mb r of
edg s be t wee
b
b bar−i · Y 1slash−bar 0 f ral 1 ≤ ≤ g.
Xbi a d Yb is a e − las t c | X
|
Fin ll−y w e m a
n
a − g a i n u s e t h ef actt a − h t den se gr phs c ntain ar em atch n s to ded
uce ha
the e i a ma tching o f s − i e at le st cN parenlef t − slash40l) b etwee
n Xi and Y i fo r a
≤ i ≤ g.
1
We a ppl y t hi s o s − e r va t i n t bG = Gb(Πm, Ξ) a nd t h
pati i − to nΠb=bΠ w1) Th us l e us c h ec−k h e r e q u i ed h yp o
e e f r u ro bse at on t
on
tha t t−h eco
ext k
par
rs t
n
u
t
l≥
c e omes in
a80ppl 03.F
/1ndi
m δ b−parenright, oewhis−l t ε≤ c
0 orres 1n − op d00 sto εb ≤δ(b)slash−parenlef t 0 o cm). 2nth−T e n wu
o
i
)
re o
g
t = dkw(1)200e
b i join ed
er , Xi
t Yb
= dδ(b)kwparenlef t−one/(
b
6
m
parenright−threee.Mo
0
g
a
tl e a
t
| Xbi· | Ybi |
c 10
≥ 10
c ( k
2 k
≥
w
˜
ed es o f G(Πwidem,
(Xw(1), Y i() of pa
)δ 2m
3
)
e
n
Gw (1)
m )2
1
1
δb) k2m
0 ·8 m3 · 4k2
( w(1
202 m
3
kw(1)
Ξ, a nd h
delta − parenlef t ≥
=3
δ(b)k2m
e n c − e t hecor r e s ondingpa
w(
b)Ui1
)
r
|W parenright − G − parenlef t − e − bar4wparenlef t−
edgesoG
0· w(1) =
·k
320m3
e − callthatever ye geof G(Πm , Ξ) c r − o re p no ds to a
(U m, W m)fro m Ξ
W e ma y d s h ala s − su me thata te st ((b/(64 0m3))k m/
s t ∈ b d
par
W
aecl, tat of ryreve (, Um
t
U
|
ed g of
goe
· 30
2k
Gw() an dsi ma l yiti oi e d
(
w3 ). 2
G
≥
300n22
me(Gw())
ereoif
Th
rt
k
)m
6δ )3·
(
he v
e(G) e( G)
0mn2 = 600k
W
ert x
00
km 1 6me
n
mn
m ya t me
t e(G/60mm
v is oin−e d to X
1sb
)
4·
≥|mn (w mδ(b )
i
y
n ed
a lea st
0
edge so f Gw (2parenright − period C le r − a ly , th e s a m e arg u m e − nt
ap le d to Y wone−parenlef t ) s howst h a − t isjo ne dt o Y 1) by a t eas t
| Y i() bar − delta(be(G)/4 · zero − one5 m n2 e dgeso
onp
s Xw ),Y w(1o
(1cn−icl fn
m(
atle
o n t− e x c epare
t o na lp ar
e
pair h s rea r e atlea est δb) | U i a · i | W iG | (G)r e(106m 4bn 2) dgeso f rGs u
a n d on ee tof e er su ch p i (risj o ined w to ev by t ls t e
t
jir)/(1
s|X iw1 | δ
δ
(b ) G54
e() 2 ≥
()Xbi|6 w(42 | e(G)=δ(b)Y|wi 6parenlef t−one )|4(
e G)
4 · 10
(1
egd 6s0 4of
m n2 − 1)), while
(brY w (1ef Gt) h
m,
mn
10 m
1
0m
i
n
G w(3)
hidgesof
et. hF i n sialy ,if u t isov
t
t h ye al ea scom m
er with
to get)h
∈ wtV(Gn )gand
v
( −f l
≥{(Xi
dL
n
1) Y wior):1 ≤≤ wi ≤eparenlef t−one−parenright
m
1
o
o
w
w e 4.Rgive aa ndoresul m
ts t ubh−a r − gat−commaa phthso ofuughl − etnif alt−i orm bipatecn−h tie − tgraical−comma
se ct−i o
ma y phsb.eo In
f i thidp en d e
int r − eperiod − t Na mely−comma we p o e tht−a , u − n e − d r q − u i te w ea k h
y o h − t e es , w ithv r
la rge prb ab li t − yarndo m i n ducd s b
vr
a oms ubg ph H i nqu e t in , w em ayalo wa n“a
ar 00to “ mark
d
spu eprex on enti laly s m all i n then f t
Hu, , ) be a nεu nf orm i p a − r te gr a p hw t h b ipart to
=V 1∪ V2 , w her bar − V 1 = | V 2 = u ≥ 1 a nd w
ε
LetH =
V (H −
it ed ged ens
e(H u 2 = . Le t d1,d 2 ≤ ub et w − o given psi ve in t ee s . N o w e lct r
a − nd o mind uce ds ug rap h of H i n the ol ow nig ma ner .F i s t anad
v e r a − sr c oh os e s a se t S1 ⊂ V 1w − it
1 S
\ 1 e u p r ob abe
1 ⊂ 1 S w i h | D1 = 1, i tha llte d1−s ub st sof
V
e
e
h
N x u de t he l kn o l d g o t e e S a d D1, u a
v esar y π c ks
d
se t S 2⊂ V 2 i h 2
≤
u l go gu, nd
we ran
o m y pick a e
D ⊂ V 2S w i h D2 | = d2, wth al t h e 2− ub et s of V 2 \ S2 equ pro b
be Let us c the ou t o m eo t he ab o ep r o d u re a r a
om (1, 2; S1,S 2) sub
ga
h o
H o s i p y a (d1 d − ub ra p h o f H.
L em
ma
11 F o revery
c − o n st a t − n 0 < β ≤ 1, t − her eex
s−i t
a cons an t 0 < ε = ε() ≤ 1 a n da
na t − u r al
num be r u0 su
h
th t − a, fo ran
y eal d ≥ 2u/εparenright−one /
and
an y gi en
g − r a p H = H(u, %, ε) a sa ov e wi h
u ≥ u0 and
% ≥ d/, th fo l − o winga se t o − in holds If d one − commad2 ≥ d r g − e
ard s s
o ft h ec
ho c esfor S1 an
S2 o f o − u r a v es−r a ry , the proba i l − i yt
H fa l
to
ont inat l eas d/2 e g i a
m
o st β.
Pro o . Giv n 0 < β ≤ 1, we ho ose ε = β2 /16 > 0 and s h − o w tha
tth
ch i c − e f − oε > 0 wil d . I nthe s e q u e , we as um e t ah u sla g
ee noug hfo o urine ualiies t ohold. Le t o ur ad ve rsary choo se the set S1 ⊂ V.
eca t at | S1 ≤ u/loglo g u. W e s ho w f i r s − t t a tthese U o ft h
ose v e r i − t ces of V
t h − a tar e ad acent t ot he ve tie−c sin o ur ra dom st
)
h n (1 − εn w i h p b b iy al ea t 1 −( 4ε) / I n o d ert o d o s ow
g ene a t e t e v rt ce of D1 on e b y on an dp oet h at, t y pi aly
n
a
, i e a t pe w e e n lrg
et e st U b y a i rn um 1 b e r of ver i es .
L et s r a ndo mly c os e a e re x va1 m on g a t h e v etic es ofV 1 \ S1t
b e h e f ir − st ver t x o fD 1.De no et by W t hes et o th e
r − et i − cs − e of V 1tha
h a e f w e t an d/2 <1 1− ε) d n ei1hb u r in V 2 . T he , b y t he hyphen−re
gu ai o (V 1, 2), we have | W | ≤ ε1If v b eo n g to W, l us sa y t at iti s
bad v 1 rt x , w h re as i velement − negationslash W l eus sa1 t h at it isa
t U ⊆ V 2bet he s etof n e g h bou so f v i n V 2. 1 i− Si mlar y , sup p o eth a − t
o rs om 2 ≤i ≤ d1 t he er tices i .. , v
ha e alr a d1b ee n p t i n o D1.W e rand om l − y i k − c a ve t ex v f ≤ im1 V1
({v, .. , v } ∪ S1)t o e thi e ith v − er texo f≤ 1an d d no e b y U − h
s e o f n ei g bo ur o f v 1 period − period, v − . T e n i f|U i− ≤( 1 − ε)u,w e le t W
b et e et of a l − l h e v e i ce sinV 1 t h at ha e f
t c si sco nine d u n
1
≤ dNow, su po s that u − rpr o e − c s ha t − er m in ated w i h a se iD
wi
t U U 1 o f c−a r n at yU | <(1 ε)u. S inc ee a c h g o dvert x nv i re
a esth s ie of h neighbou hro o d f D1 i nV 2 b yatl e a t dε 2 t e u m
berof go o e
ele m ntsin D1 mu s tb eles t han 2u /(d) ≤ d1/2.He nc e atle as
h ao
al h ee e me nt
m l er ha d/2dε d ≤ (4ε ) T h uU
≥( 1 −ε )n wt h pr aob b i t a
s,| |
1−
(4ε)
.
leas
≤ d1 s su men ow th at o r p rc es ha st e r minat d wi h a set 1
w i
h U = U o a r − di lit y U| ≥ (1 − εu. W e n w le to radve s a y
pi c kh is s e
S 2 ⊂ V 2.T hen h e pr a b i y h2 t tle s t
alf o 2t2 he d2 ≥ dv tficeso D
s hou ld l e o u sde U i s a2 m dtwo − st 22 (εn/d n− ||) ) , wh i h − c,f os
u f f icie n t
la r e uis l e ss
/ ewe
T hu , the r oba blt y thatou rra n dom (d1, d2−
H c n−t
f
d
d u 2 ra p h/
d
s
u
t an d2 edg e i b o d e d r ma boe b y (ε) / + (8ε) 2 ≤ 16)
β,
5
A
sr − equ ie − rd .
=
su fficien tco
nd i ti o n f o r
u ni fo
thl−p a r − ti ionV (G) = V 1 ∪ .. ∪ V (l ≥).R ec al lt
hat Gi hyphen − epsilonu − n for m if a
pair s (V, V j ) (1≤ i < j ≤ l) a e ε e g uar. Mo re o v er−comma o bs e r vet h t i
n ord
e
t oc he k th εhyphen − re g
j
e
b a ra th er si mp c o d iion im
t r so u that the ε−
r gu art of (V, )i sim p ed
p o ed u p n t n er ec t n fthe n egh b ur h odso ofp a i r o ve r ces T h s d
a h sa b e n e x p o it edi n mny plac s s ee e g . A on , Duk L f m a n n R d
la n dY u t er[ AD L RY 9, F rak R o dla n d W isl o n [ F W 8 8 an T
om son [Th 87]( s e als [ T h 8b] ). h e ol w n fc isaslig
ef i ne me n o e r le r es uts in [T h 7a] an d [F R W 8 8
L e
m ma 12 . L t
G b ea d− regular bip rt teg ap hwithbipar−t i io n
V (G) = V 1 ∪ V2 , whe re bar − V 1 |=| V 2 |= n a nd
d = pn(< p ≤ 1). A ssu
m eth t − a for
sub s − e t B of V 2 wi h b v er ics−e and som
e ε > 0 weha ve
(1)
| N (x) ∩ arrowdblright − parenlef tunionmulti − x)T −bar Jproportional − lessequalparenlef t − F 1plus −
n
o v e a u no r e ed pa r s x−commax ∈ B
e
T h en , o ev e y s u s − e A o V 1 wti h a v r tce , w e a v
w−he r the
s u mi sa−t k
(
)− a
bp
≤
εa( n−a ) p2+ a
p
ithx
6= x.
i=1
F urtherm
t wo w i − t h
or , u ing ( 1 ,
bot e dn sin B
−)np ≥
oun ti g d − ire ted
e ad s o
X
d(i − 1) +
P
= a + 1di = e(V1
P
dd( i−)1 =
P
Since
i = 1 i = e A,B ) and
e(A, B), b the C u h y − endash S hw
parenlef t − one
+
+n −
wh ch , aftr
b− e
A
B
−eAB
)(
eem en ar ycalc u io s ,
−n
g th
\ A, B) = db −
− 1)np ≥ ae (A, B)((A B) −
ε)b
a
e−l n
p aths of
a)
+a
m
abpn(n − a)(1 − p) + epsilon − ab (b − 1)p n − two(n − a
naol
e
ll
wss
m ma
e
uffi Ac e antc o mndtonaf o−r th neη q−unif
o r mty o a aregula ub
. Let
b a d− r eg
a rb pat
g pa h w ih ip
tion
V
V 1 ∪ V − two, wher
eh
ve
∞ up
o Pc
| V 1 |=| 2 |= n, d = p
o
0 2
n
33
xcomma − x0
<
p
n
o w
2
∪
N
0
he e h 0 mi sake
nd
,
0
nov 2 ra luno dere d13 a r x, x ∈ V 2
wi th
x 6= x
and
N x)∩⇒
1
any
ditin t v eti nsx, x0 ∈ B, a nd 0B00 r f or mth
s c r c e 0 x, ∈ 2V
e0m e l 2
s u o vr dal u n r − dei
r e is o f d
d
≤
|N (x)
(
c
n − e dhe c
a
≤ εabp2n
, e n t i onm p − of de s ty p do (comma−f i,
) = 4(A,
∩⇒x) J− ∝ F
+ε) b
p2 +
4
y − b∞ e mma none − two, ∞
t o − parenright − slash
,
ar
f
e
ug
o
r
b − ap)r eh t e d(V c c − one − o − comma
two − d) =
(
V
Hence
bar − d (A, B)
−
d(V, V ) =
4e
12 | bp
qu
red
asr
L e A be a s ub s − e t o n] =
i
wh e r ebo= t hG
ce
A parenlef t − na nd s ta e
6 . D iffe re nce gr
a ph .
{01, ..., n − 1 n V (GA ) = V ∪ W,
op i eso
v∈
V an
w
er
fco
m on n eg hb o r s o f o di
i e i e s and v th
, 0d thatthe num
b
a e
y
a tb e o n g tot h e s m ls
o th e b pa t ti n d e pe nd son
o n0 t h
l
e
v
n
m
val u of v − . I 0 a t t s vaue is t 0 a m a t h 0 n u
b e r tA( v −) o o
d ed p a r (a) ∈ A × A s u − c h th t a − a ≡v −v ( m d n.hT
e e o th e s r uct reof GA s lo el yrea td to the beh a iou rof t h en m b est A(
(1 ≤ j < n Ou rne xtr e s ltdels wth the di t i bui n o t h tR(j) f r a n
do mse t R ∈ Rn , p) . nt h eseq u e , w w rite ⊕ a d  f or ad di o n
n
sub r a c o m od uo n rps ec tiv ly.
Le mm a
14. F o r e − v r f i xed
0 < ε ≤ 1na d 0 < η ≤ 1th r − e e exists
co nstant
C = Cparenlef t−epsilon , η) for
w hich t e − h f − o low i ngh
ods . Fo r
e ery p = p(n) ≥ C/ n, he
rob−a bl t − i y that R ∈ Rn, p)
sat sfies
X
(3)
{t Rj) − ( 1+)n2} ≤ η2p2
(
dsto 1
as
n→∞ .
e
al
1
j
<
uh
h
R
j
)≥
)
n
P
oo . F or i
∈ n],le I b et e c h r
act e is R g
u − s d i i − v de
m
e s e o f alh n
t
pa r − s(i ,a ⊕
cass s
B1()B
(j) a d
B3(j) i
su c
a w ayth t
bn/3c ≤
B
(j |≤ dn/3
|
rn gi whic ht he siz
or a l l ∈ {1 2 },i f j1j+2
s of a n tw o e r b
o t h e Bj
t a,
= n l h nt h e se
u
of no rde r
pa i s t hta n tu r l yl c orr e pon dtot h e e m e t of B (j 1sh o uldbe
t he sam asth co r s po dinsg e t for B(j2) Now fora l 1 ≤j< nan d
∈,{ 2, 3 d ef in t h era nd om v a ab l s X (jl) anP d X(j, l b y et ting
)
X(jl) =
IIij,
(ii ⊕ j)element − Bl()
⊕
an d
{0
f X(jl)
<(
1ε+ )Bj | p2
,
Xparenlef t − j, l) t − o he ws
X b, l =
j − commal) X(, j
r−e−h
, l)i − s
e
t
2
| Bl( j) | p }.
X(
sat
m o 3n − summationdisplay − minus1
− 1(j, )(1
bXj(, l
+ ε) | B( j |p ).
l=
1
H en c e i is e n o g to
as n → ∞ we h a v e
1bX(j,l ≤
=
ηn2p
e cl s − etwo − B | parenlef t − j ),
m
B
=
(j ) a nd
t o
.
ma−t ef i s tth eex pe ta ti n E Xbj − parenlef t, l)o fXb(jl) N
e
in
1
s ho w t h at , w − iht p rb b − ai lti y ten ng
Z=
L e t us sti
X,l i a su
j
o−t t
a
ε)p
+
b m
EX
X
(jl))=
(m)
r
r
2m − r
2r
p
− p)
(1
r = r0m
= mp2
X
m
−
1
2r − 1
−
m2
p
m − summationdisplay −
= m = b − greaterequal
( m
−
zero − r 1; m
)
)m − r
p
r − equalr
=
(
r−1
1
− )
2
− 1, p2 ),
(5
( Se ,
rc a ,
th , 0a nd k0sch t h a(
≥ (m p + k m ,p ) ≤ e x p − 3 mp0 .
fornstan c S e ton 6 i n Mc Diarmi d [ Mc D 89 ] ) Th u s − comma in o u
e i e c ( 2 2) . { }
b≥ (r0 − 1; m − 1, p) ≤ ex p − (εpm − 1) ≤ e xp − 1ε
we a − s su e − m t h a − tn i s al gee two − nuf ogh r a l
h . e e , )≤ 3n(−1p x εnp 0) a { d co e }
p .
b
E () = E l =1 j = 1X(l)
)≤ np2 e x − 1zero − one
εn
e
u al
q
np →
n
→ ∞,
yM h−ten arkov the ’ s i − ri − n gh t - h a n
ittha
y i−s to( n2p ) .∞T haserf −eo r − e, b
a
nt h − i s a − c e m hple t ro bð i , t
t−h
co
p = p() > C / n for s o m e a rg C > 0 u s , n2 < o ogl gn .T h u
we h ne o rtha ss u me th a p = p( n) saisf i esth es c n doti n s.
I n t he r
P
m ain i g , 3 t heP ne− c 1bcal pa rt o t h e pr o of w e h lc o mpute the v aan
p
T hsideu−s ,i
of the = p(nabveissuh − ci
of Z
=
l = 1 j=
ar b eis co ncn ta te ar und i e xp ecat on th oug h a d i rc a pp i ati o
C h e b s e quoteright − sn − i e qua lty l b
Frs t,le t sn ott h atw i m =| B ()| asb e fœ, the vra ianc Va r (X j, l
ofbX
m
X
g
(l)isno 2m) r2r e−a
ta2nm−
etr
(,
r
requal − r0
mP
(m)
2r
r
2
(1 −p
p
m
r − minus b
= r = rr (r −
= m( −
=
m
m
4
1
r
−
m − summationdisplay 2
−)pb ≥ parenlef t − r0 −
n
) r
2m −
p
−
2
2) + mp2 b ≥ (
11 np
=
( .
sup p osea
d
lso
th a
i fj 1 +
j2 =
W
=
and
C
U
2
=X
(j2, l .
=
parenlef t − vU 1, U 2
)
We
v e C − o(Xb (j, l
ha
,bX(
2l
))
=
2
E (U 1U 2)− E ( 1 E ( U2 parenright − period e − W sh llfi rst estima
efrom a b o v ethe v a l u − e o
E (1U − two). R e al t a − h t w e w tre R fo a r − ad − n o m e
el m e n o our p r − o b ilP
t sp ac e R n, p ). W e h ve
( 6 ) E (U U ) =
U (R U (R) P o b (R = R ).
0
1
2
R0 ⊆ [
]
1
) r
0
n
u
Le t s s ythat R ∈ R − parenlef tnp) i s x ept o al if et−i er U (R o
U 2R) isat le a as l − ag − r ea s um a − x = b lo
gr niminus−parenlef t 1two−slash ) l lg n − period nc e U1 y an d U 2 areat mo t n,
go g n t
o ra tt e n ton o n o - e x e pt iona l R∈ R (np ).
Le u sint do uc e s o m e n t@ o n. L l t Gs (s ∈ {1 2 }) b eth e drc t e
g ra p h w th vertexs e t bracketlef t − n] an de d g e se t B sj. Th s e a ch G s
s im ply c on st
of s o meis lted e dge s . M r eo v ,t he d re ted g r a h G1 ∪ G 2 c o nai
n n
cy cle o f le ng th 2. L e t H s = Hs − parenlef tR ) s ∈ {1, 2} )be h e r nd
o ms u bgr ap of Gs i n duc ed byt he e lemen t o f R ∈ R(
1(R
W e no w c ons de r t h e
∪
H
2(R )
⊆G 1∪
s r u tur eo f H.
L
G.
e
u
t
sayt h
a R ∈
R(n p)
i t pyic i H
= H (R ) i − s a m at i − hng.T 3/2e p − radicalrb abili ty t a t
R∈ R n − parenlef tp) isn o ty ca s a t m o s t n3 ≤ (gl g n ) / n. T
u s, t h c2 nti uti ono 3/hy−p
no n2ex e − cp − ti na3 2R0−/2 n] i n 6 i
e−1/ i a − l,
n
t − a m o st u ma (lg o − g
2n = O(parenlef t−l o g ) ( og o n) /n
)
∈R (np) t h ata r et yπ a an d n o −xc ep io n a l T h s H = H(R) wlil
a w y c ns s to f so at ed e gs.
Tb o u nd h c o n r i u t o n o f − th e y − t pi a l − comman o n
e x e − cp t i o a R0 ⊆ [n]t
t he sum in (6 ),we s al n e d he f ol ow n g o n s uqen c e o fal
a r gedve i a t on
ine q u l i t y o f Jan s o n , Lucza k, an d R − uciń ki [ J LR 0 . Lt J b e ag
aph wi t m px − i m a l d gre e at ms t 2 , it h m e d ges , a ndn 2 e − vric
s of eg r−e e tw o Le J bea r a n d o m i nd u c d sub
d mlya n dn dep e n dten y, e a c hw ih pr o abl i y p. The n e np
s
tP
ha th ep rob b l y
q u lt of Ja osn, L cz ka nd Ru 2 c ńs ki gi3
h
a
i
t tJ h
sn ed g e ss a t mostex {− mp + 2n2p }.
W e renow r e dy t ob ou nd E0(U − oneU 2) = U 1R0) U 2(R)P rob (R= R0
w he e the sum is take n ov e − r al lt p i l − a a nd n o n e − xc ep tio n
al R0⊆ [n]. F
p 1 2exp − (m
X
p)m − u1m
xu ( m2)p2 21
2
≤
X
mxu
(m 1)p2u (1 −
×e
2
1p − parenlef t
−p2 )m2 − u
4
matwo − x + np + 2np3)
f − oe
(
1+e (
)E(U
2
2 ≤ n minus − parenlef tslash − one2) og
(n− /2( lo−g log
+O
n
s−comma
T hu
X X
3
n
1summationdisplay − threen − 1
)
j)
=
−1/2 1= 3 =1 2 1 three − slash2
h 5 ) c t b ≥ r( −1; m − minuscomma − one2 ≥ (m − 1parenlef t − r −
zero parenlef t − parenright − one − 2) m−r−zero ≥ e x p {−3n p2} 1
ht @
X
X
−1 b 2
l = j1
2
n
prob ab i − l i
Z
≤
2 E Z) ≤
h
n2p2 e x ( ε
o ds w
10ε −lo g ( 2
np 21)
η
Lem m a s 13 an d 1 4 im m e − dt − ae yi mply
− i−f r
o m u b se R ∈ R( n, p) i s ηun
o
m
y
p
h − eren
sa
od
dnat u a n m b r
Fllo win gR u saa nd
Se
r ´
d−e i [ R
e
Sz 7
(see
¨ o[E − F R 86] a nd G rh a m a n
alo E r d o s F an l − a nd R d −
7
d o¨d l[ R − G
8
e
w e i n rtoduce a ga p h G(n,S )th a tre f lct
st e ar t m i
1 2 V 3 o t h est[n]a d ,j} i an e ge of G(nS ) f nd i o n e
thef o o wn g th ee c on d iio sho ds :
( i )i ∈ V 1, j ∈ V 2 and j = i ⊕ k f rso me k ∈ S ,
(i−i )i ∈ V 2j ∈ V and j = i ⊕ k forsom e k ∈ S,
( i − parenrighti ∈ V 1, j ∈ V 3 an d j = i ⊕ 2k for som e k ∈ S;
h e re nd b elo w ⊕ an d  s t a n df or d dt o nand s bt act on m
o duo n Cl early , if k ∈ S t hen h ee ic es i
r i a ng l − e in GnS). A r na g e o G(n, S) f t s − i
e a e i ter s e di nth n o −nt − r v al , or sp n a e − no
n c e th e yr e fle c th ear t mh e t − i t − s ruc ureo f S n
be ow
Cl a l e a h G(nS) co nt an s p ec le y nStr vi l rt i n g e an d inf a
)
G(n,S sit h e e g e d − i o int un o n o f t o s t r i n g es. Mo rei m po ta n lt, yt
h
e
nu mb r of s p o n ne o us ria n gls
i n G(, S ) depends n t h enu m
brofrt
me
ctr pe s i n S,t hat iistrpl e so f dist n c e em e nt a, b, ∈ Ss uch t h
ac
egra ph G(n, S) o n t n s
vetice i ∈ V 1i ⊕ a∈ V2a nd
neo u s r an g isa s s o ciaed
h
wh
e
et S
c ont anis an ar t mti tr
a
t eh a soci ted s po n a ne o u srt gian le wt h
i⊕ a ⊕b ∈ V 3 a n d , o nsver el an y pson a
wth s uc h a p ai (∆, ) T hus,i n od e rt ove iy
pel
iti
e nuo g h
t o l o of
or a p o t n ous t i ang
iG(nS )
N a tu a l y w e sha ll be p ar icular l y n e restd inth e sctru t re o f G(n,R
f ra ar n d m su bs e R o f [ ] I w lb e atercru ci t h t , o a large ran
y
p caly uni fo r m an d h en e s p
d o se t R ⊆ [n], t h g aph G(n R) is
t
w
r
s
ase a s ho
n by ou net tw o re u lt .I nt h s qu eli t wil b e c o nvenent to ex ten
t h d fi n
such t hat Gn,S ) i η u nf o m fo rs om e 0 < η ≤1 , t he nl e u s
a y t a S i sef isη un if or . M o r v , g ve n b> 2 a n d 0 < η≤ 1,
w ed efin e h
s
n o n f ( , η) spa
n s f o S a beov i nt h a na l ou w ay.
r
F ac t 16 . F
o
e − v e ry 0
n t − s ant
C = C(η) suc th a − t,
b i l − i t yt hat R ∈ R(np − comma) i
n → ∞ alo−n g the dd
nt g − e e
< η ≤ 1thee−r
xi t − s a
co
√e
if p = p(n) ≥ C/ n, t en the pr o a − b
s ηhyphen − ui − n orm t ends to 1 s
r − s.
yd d u ce h eo l ow in
r m t y a d spa sen e so R M∈ R( n M r M .
F − ac t 1 7 . For
eve ry 0 < η ≤ 1 t h er e x − ei − s t √ac n − o st
n − tC = parenlef t − Cη) suc that , if
M = M (n) ≥ C n, t h − e
nthe
p o − r ba bl i − tythata R ∈ R(n, M ) i
−nif m
a d (4, η− p a s tend to 1 as n→ ∞ al on g
theo
dd i n tg rs
Pr oof W e s t − a rt by nt−o c − i g h − ta t − comma f ora n y 0 < η
unio r m i mpi sth f a G s s ya ( , η)− s tarset. h reforew a p o e G dt show
ηtha f r n 0 < η t≤ 1i − f,C n1/2 ≤ M p = M (Tn)≤ n f o o msuffi
ce nt lrg t econ tnt C th e n G, = G(n, R) s η n o m wi h p oba bie
ty 1i−
n → ∞ o n gth e od d int eg2 s .
P ick η1 = η0/ 6a nd ε = η/ 3 Le C = (√ 1 + ε) C 1, w e e C − one =
C η1 isa s gie nb y F a c − t 1comma − six a n d a su m e t h@ Cn ≤
M = M (n) ≤ n.Se
p = pn ) = M/( +ε )n W e m a g ne ae R M ∈ R n, M ) b y
pic k in Rp ∈ R(n, pco nd i o − nde on Rps a it s f y ng | Rp |≤ M, a n dt h e nb
y a d d i n r n o e l emen tsf [n]R pto Rp
ih p r b a l t y 1 − o1 a n → ∞, w h e Rp ≥( 1 −ε)p nw e
h l l a s m ethat ur Rp d e s ti f ythsco d ito .A − s s m a o a t
G = ( Rp s 1 u n o m a nd r c a l ta by F act 16th se n a s o
h d sw ih pr o a b t 1 − (1).It n o w su f f i t o soh w ha , u n t
h etw oc o ndti o o n R t hs et RM i s0 un i m wht æ v r e em n s w
e rae dd e d Rp t
ge nr@e
W ri t GM fo rG(n , RM )
t
G
V
M ) b et w d soij s e co
(
of G M wi h| U |,W |≥ η1 n.
hen (1− 2ε)M ≤ %p ≤ %M a dt
an d
R
M.
Gp f o r G
t ane
n
comma − Rp. Le t U, W
w di t − in t − c
di nt
v e tex
u t M = M/n a d %p= | Rp /n.N o t − e
@| %p − p| ≤ε p. M r eo v , we ha ve
e
GM
)
≥ Gp( W
≥ ( − )%
an
|
U
W ≥ (1
(U, W
U·
W
d
e(, W ) ≤ e
G(U,
W)
G
≤ parenlef t − one + 2η1) | U | · | W
) = 3G
+ 2pnU
ε
≤ (1 + 2η1%M | U | · | W
.
M (U,
W
M
M
U
|
p
No
dGM (U, W
1%
M
⊂
cass e
th at
utM
R ) s ind eed η− uni o rm , as requ
I − n he sequ e , itwl−i b − e n ece sar y fo r usto view R n − parenlef t, M )a s re em b l − i ng
˜
0 = M/m. W − etoR−e−c−a−p−s−e−h−t−e−n−f i−e−d he ae . A u m e t h wide
( e m , Mzero−parenright as th e
i
˜ R n, m, M 0a rreeq up robat be , n
Th us al −m tple s e =∈ Rwide
R
˜
Ri−parenright = 1 ∈ Rwide (n m, M 0) 7→ i = 1Ri ∈ R(nM ) s me u − s
r e − hyphen p rese v − r in g − period W
sha l − l a so c onsid r h − t e p ob abil−i ty
)
˜
e
p − sa c − e G = Gwiden, m, M 0
f h e ba l − ac − ne
3 - eco m − po sa e − lm − hyphene g − de colo redgr p h
e = parenlef t − tildewide − Gn, Re)parenlef t − equalG)m = 1 d ter mi n − e
G
b yt e G = G(nR) 1 ≤≤ i m), w h e − r e R = (Ri)1isai rand om
l − ee me n
of R(nm, M 0). n h sspa c − ee − w c nsi er the event
A(b − commaη) t ata g ap
fr o m G s oud b − eb − parenlef t, η − res, nd d n − o te he co
e byc ndit i − o nin g on
dtional probabilt−i y p − ac ob t in e − d fro m G
b e
A(,
y G( n − commacomma − mM 0 | b − commaη).
)
b
0
T hen teh e exst co n s a tn , sk0 < ε = ε(m4ξ)ξ ≤ 11 a n nd d C <=
C(m, kb,,ξ,δ f o w h ih te
f
ol wing
hold fo
anys
uffi cie tylar g n .
Su
ppos
Cnone − slash2 ≤ M = mcM 0 h = m l M0 (n) ≤ n/ ( ogl g n)2,
˜
e = ( Gi) = m1 ∈ Gwiden,
e 1,. .Π
e m beth e
l e−t G
me, M 0 | b, η an
dlet Π
(ε, k0parenright − hyphen c noni ca se u en c eo pa r i t o n
th G, w ere as u a ε =ε (, , , k − zero) . T e t h pr o a i − b t t h tt hre eex
o
ists a (δ, k0, ε; G)− flo werw h c h
t a i − ns n o s o − p n
a − n eo s − utr
n
an g − l − esis s ma l e th a − nξM.
et
roo . ut β = ξzero − one8mδ − 2,a nd
0 < ε = ε() = ε(mξ, δ) ≤
be Pas gi v e n b yL m m a 1 1 .
F u r ther morel l e − t η a d
K 0 ≥ 1/2ε) b es uc tha tL emm a 5 olds a d e t C1 = C(η be a sin
Fact 17 . F n (a−ll comma − y set C =
√
m a x {C, 1 07 m K0/(δε)}.
comma − epsilonη an
Wes ha l − l s ho wthatforsuc h a
choic
of
C t ha
˜
h t n i s uf f ic e l lrge w h ev e iti s e d d wide
m L etu
s rs resae o u esul tinterm o f m co o r − ed g p h s G =( Gii =
fr o mG(n, mM 0parenright − period e t B( δ)b e he e et t h@Gwide
˜ s h
s
ou d b e b η)p
r
e
andm o r eov−es−i t h ou l − d c on a − t i na (δ, kzero−comma ε; G)−
flow erwi t
out a sp n a n e − o u t ran g l − e. Weh v − a eto ho w th t
robB(δ) |
A(, η)) ≤
( ξ
Su
p o s w sh o
tha
(7
Pr b(B( δ)) ≤ ξM/2.
as
= P r ob (A(bη))
˜
L tu sfir t enestm at et hep oba biity thatafi x d (, Gwide−
ow
sp o taneo u s
(i−parenright|=|Y (i bar−parenright≥ ),
i :1≤
ev e te x v ∈ V (G ), a n d
(
the
f amil{(Xi ,1Y
e rcont a i n n
i≤(g2}
w ee
˜
wide
s−s
o
h
io of G, a
d≤i u ≤ g0 = ndgf/e)
aa e ,c ntainedr. i Wn asn a l vert ex lassofth eti lpati
sar allth2 e Y () (1 ≤ i ≤ g). Blo w we s h a l o l c o n d r X parenlef t − i)
an Y i − parenright f or 1 ≤ i ≤ g0 Al o , l etus m e nt f c m l pl nt ne ss t hat G
Eparenlef t − G) |= 3nM = 3m − nM
0
e − dg e
Sup p se nowt h at t he Ri(1 ≤i ()w((i) ) h veb en c0o e n . Th u, i p r it u
a , h eg r a phs Hi = Gw(1X ,Y ](1 ≤ i(i)g) ni) m el yth
b i − pa rt tes u bg rap s o f G(w ) w th b pa r t on V ( Hi) =X ∪Y ( a n
d ed g
s e t EGw parenright − parenlef tX iY ), ha veb ee fix e d . B y t e
definit o − i no fa (comma − G) − f l we
a
ea ch Hi(1 ≤ i ≤ g) is ε − nu i − f o m − r
ndf r l 1 ≤
,
a
r t
w(one − parenright(X(i, Y i) ≥ %0u2 wh
er
%
= 0 −6 δM
0/n.
y−d
Let u no wpick R w2( a , n − d s t u
t h e n e g h ou h
oo Nw ()parenlef t − v) ⊂ X( (1 ≤ i ≤ g0) o − f t he vrtex v in theg r
aph Gw2 in s e d t ese2 t X (). Pu
˜ G −f lower, w ek no w
d =%() 0u. A g ani b y thed ef initi onof a (δ , wide
)
)
that(i−parenright d(1 = | w(2)v ) ≥ d = %0u N w le u s c ondt io o n
use so fca rdint iy d(i m of t sh rs t xX ()\ o S(i − parenright v a yr1 e uai l y li k
t
l t h c oh e an s N () v), wh ee S parenlef t − i is he ne hbo 1 u ho o
ed f qv wt ne yXS i − parenlef t)i
e grap h
w( ) Gk. Fur 1the mo e , w iem aket h si pleb utim po n
h
o bs e r v a t no t ta , be c use wec ho
S se Rw(2) r an o m ya nd ni orm
ly f o m a ht M0 s − ubse s of [] \
1 ≤ k < w 2)R, a nd bec ausew e
(
ah e e − d ide d o nth cardin lt es d1 (1 ≤ i g0 ina dva e , h es t
N ( ) v) ( ≤ i ≤ g0 ar allse e − c t din d ep e n d n − et − l y ) w ) )
pick Supp R
n
w
se(w . oLet
( wparenlef t−two) ≤
< three−parenlef t)) hav
(3− ne−ighb el
i)b et ihe wGw
tN w(3(v (htth eRi
) ⊂Y (
− asoburh eenood choo−sen
v w , anthi
Y. ut )d(i) =| N ()v) | (1 ≤ i ≤ g), a ndnotet at agai n d(i)≥ f d = % u W
eno Pw c nd i o wo 3n(t e v al es ot h e d(i − parenright(1 ≤ ih ≤ g) A 2s b
oeun0 do it n h
isond t onn g ,f o rev ry 1 ≤ i ≤ g, l t h e sb s t s o card i nlty 2 o
emm a 1 1 o allt he H (1 ≤ i ≤ g = %
%0 = n /c e a nda t () ≥ d(≤j i∈ ≤{ 1, 2}. A lso , by t ec) hu o ie o f iC, wet
l aav d ≥ 2(u/ε)1/2. Mo eo v , sine = mM 0 ≤ n/( oglog n)2 , or rg e n ou g
n w e h a − ve | Sj() ≤ u/og lo g u f o a ll 1 ≤ i ≤ g0, j ∈ { 1, 2}
h the es N i − parenright(v(1 ≤ i ≤ g0) ae a l s e ec e d idep e nden l , sar
t a t ) t w()) 0 r l l t n t y a
a t h e N wparenlef t − three) v ) ( ≤ 0 ≤ ) T h u s , p − p yi gL
e mm a11 s m u a o slyt a t h e H i ( ≤i ≤ g) w es e t a td 0 o b
a − twob M i t y t a t we do no tha v e
o Now , tuso t o pm et eh Gep r oo , it s u βf f i e ≤s o .s m a t e t h nu mbe
e f l − hyphen) ow e s
rof a pos sbl can d − i da t e s f r comma − deltaG
i no r (b, ηparenright − hyphen sp r − ase m− edg e - ol ure
e
e
g aph G.C(
y , t − h e e − vr − tex v can beselect dinat m
m l − parenrightbig a
o t n w ays , the e − r a − r
ei
m pos
˜
ib1 cf chLicem sma or t5 )ahn ndgd ≤ic e
atmo sinces − t|Πwone−parenlef t wide
3 )≤
|≤K 0 +
sw (1) , w()a nd w(3)a n
umb−er
o f po s − s iblec h o − ic sfort he s t − e of p i r {(Xi), Y (i −
parenright) : 1 ≤ i ≤ g} c a be esti ma t − e dvery gen r ou l − sy√ by K0 ×
Kzero − exclam ≤ (K0 + parenright − one!.T h u − s, i − snce M grow
toin f i − ni y at lea sta sfasta s n, w e ha ve
P robB()) ≤
w
he n ver
n
si rg e n
nm3(K
0+1
!ξ2M ≤ ξ
M/2
ou g . Thus 7 ) hod s na d Lem m a 1 8 i sp o ed .
la e h ( r v
A sa n a − l mos t m meda te cn e − sq e nc e o − f te−h a b o v e em ma and
e m a 10 ew ge t th e f ollo w ng res u lt w − hic hw l − il b
are η)M − s u b es s Ru n of n.pr C ob y, Ry n,
obta−i ned f r o m R(n, M ) by cond t io i − n ng o nth eevnt h a t G(n, R) s − h
(b η) − sarse .Lt h ea socia td pr b a b ii t yspa ceof t e GnR)(R
Rn M b, η) b de no e b y G( n, M | b,η ) T h u to p c k a nele me t
(
o m Gn, M bη) w es im py ge nera R ∈ Rn, M | b, ηa n d le G= G n,R ).
se
S up p o t eint g e r m i v d e s M. C la y , s e l g r hs om
G( n, M |b,) a r e n e at d b y M − subs ts o f[ n] an d e ach s u h s u b et
ca b ed e co m pose int o m u bs te so f izeM 0 = M/ m n t h e a m
˜
e n u be o w a wide
n e c a n ge nerte a el e m e t Gn, M | bη) by c
ho i n ga g a fro m Gncomma − m , M 0 | η) and ig orn g te c o i − r n
g o−f t e d g .
N ow ou rnex tre ul t can b s − ta t eda f − so l − o w s .
L−e m
ma
9 . F or
every
b ≥ 4 a nd 0 < ξ ≤ , th re e i t − s
c nstant s 0 < ηb, ξ) ≤ 1, C = C − parenlef t, ξ), a n d
N = N ( , b) uc
t
at , fore ery n ≥ N
d
0<
ε(, xi−commaδ ≤ 1eb s−a in
b
b
O
a g ve nb yL emma 1 , a
y Le mm a 1
A salwa y − s,
bsr
eta
,
e au e o
t
≤
ξ
1 b eg iv
eL m m−a18.
ndl t C =
we
e
≤
1
)
C(m, k0 , b − comma, ) be s − a g ve n
Fi a−n l ly , l et 0
< = η(,
s − a s u me th a − t
n i ssu
m
nd δ,
c
o
eo f
,k
,a
v
y l em
e
e
t
,
M 0 , ) co n an s n parenlef t − M ε , , ε, 0k)0)f low e r d ,
m
8
u o
yLem a
w i h p r b b lt a leat1 − vre y s u h f l o e m
te
c na
s
1
b
n a sp nt ne o ut range T us t e p r
b i t th t a n e em M n tof G(n m, M | b,
a
˜ n o ust r an ge i sm a ll r t ha n ξ. Asa lr ad y me n
c o ntins ns po n wide
egraphs f rom G(n,m , M 0 |b η) na u r y co es p n o e l e met s f o m
G(nmM | bη) an d hen ce e m m a 1 fo l ow s.
We can now f i naly r o − v eT eorem 1 .
P ro of o T he ore m 1 . C ear y , t uffi cestop ov t h a fo ra y g
ve 0 < α ≤ 1 ther eisasu ta be ch o cfo irC s = C ( uc hthat t, n i
, i i e α) s
() i − f Cn/2 ≤ M = M n)≤n, t h n
limn→ ∞
the l m is tak en ao no dd v l
s u m e t a t n i s u f f i c i l y large hw he nevr e i hi nn eed edF do Arc on
v − eonienc t u s a y h that sa p ope t t yh olds a lmo t u ly t i ith lds
t e ing to 1 sa t ed stoi f in i − ty ao−l n g odd n t − e ger s .
eta cons an t 0 < α ≤ 1 b e g − ie − v n , an da su
ot t atthe H e a h - Bo w – z e me r´diresu 2 m e t on e di S − ec t
n 0im pi
that a n yse t A ⊂ [n − bracketrightwi t h| A| ≥n /l − parenlef togog )n co nta i − n s 3te r marth m
p rg re s o n p o ide d ni2 su f f ic e nty l rge . T h u s w ma an d shal as sum
th at αM ≤ n/ l g l og n) ,s icn eo t herw s e R → α3 f
(,
|= M.
0
0
0
Pu α = α/. S et b = 6α ≥ 4 an d 0
m = mb) ≥ 3 e a sg i v n i n L m m a 1 0
ξ
an d C 1 = C (b, )b e a s iv 0ni n L e m m
sgve n a t 7 . W e et C = C ( α) = max
wt h a
< ξ = α/ 4 ≤ 1 a nd le
M o reo ver ,l t 0 < η = η( b,) ≤
a1 , a0d et C2 = C(η) be a
{(4 3α) C1,C }. W e w h o
(†)h
O
st
su re
rf i
u
h
s
st
h
i m
e
t o
rop
v−e
fy
t h at
r
e yha t a n su bs tA
t
⊆
R
∈
( n
R w iha le
0
a
m
sα|0R| e
m
et
t
i s a n a t m t − ic ri l − p e . F o sm p icity, eltuswie R
a
th 0 s po pe ty.
p
L e t u0 sa r t b − y
k − c i 0g an nte g er
0
M =M n) of msuc
enso
that (α
/4 ) M ≤
M
≤
αM
→ α03if R0 ha
mu l t ip−l e
ht e
ξ M. Tuh s , t he p ro M 0bilit y t h at b, η
f
η
spa ses u bse s A M 0(nn ) i h M 0
e m e ts t h a t on o
tri
o n i na i h m tic e s t o t M 0 .
bb
e
isfr e −
subs epro
AiDt bewy tha tsw hiac tms tthe M 0ve−l
Lt
ne tha
t
me n
r
nme
ht
ethRn,M )sof ario ld
th
)
ξM 0(M n0
co neta
c inr
(nM
−
b, ) spars
iae
pl . ηT h
−M
− 0M 0
Dh
tR∈
ol ds is
nth
)(nM ) − 1≤ (ξen
(
) M 0M
)M 0
M
(4eξ) M 0
0
n
α
L e − t n w S b e e − h ventthat
∈ R(n, M )i − s 4, ηhyphen − parenrights−p a rs . T hen, byFa t 1
S hold s a − l most u − s ely . W e no wnot
3Sn − ic
m
Drf
d
s − arrowrightalpha − t
nl − M
s − sb, ee y
alpha − prime − a T a − n
s − S − o,
R ∈ R − parenlef t
)
R→
m
N o w ec alt hat n is o d a nd w r − i − t e n = 2k +1 . O b e − srve th ati f A
as ubs e tof R w it h a tl ast αR | e − l e m e n s t hen at leaso neof t
h s 0b − u s e A 1 = A∩ ⇒- .element−period . k mapsto−arrowdbllef t a − bracelef tn d
A2 = A ∩ ⇒-, element − periodtriangle − period,mapsto−arrowdbllef tk {} m u s
h v e a te a t α| R |= α | bar − slash2l e me n ts a n dt h t − aA i(i ∈ {1,2 }) c n
ai nsa na rth me ic t r − ip l e a d o ly i t on ta n − s a n a r t − ihm e ip o g
esso n ofl en
osh o da n d T h o em 1 i pr o e .
Cor la y 3 m y bed e u c ed from T h e orem 1 na r utn e man nr .
o lr a d e
Sk e tch of th e pr o−o o f C o r o lar y 3 Le t s = s(n), g = gn)
an d α b e as inth est a − t e men tofo u co o − rllary.P ic k C0s − uf f i i − cent
ylar e s
tha tw i − t h p = p(n) = C0n −/2 n − a d
hT he ropro a iit yh at )af i x edst a sin ) o ft h ecorola rymee t
o e t h a n e l m e n i s Os n − . H e n c e t h e xp ct e n u m b e r
s
o
c h s et i s (s) = ( n√ . Tep r o b a lty t h @a f i x − ed s t − e − radical
u
as i − n(ii ) m eets R i
t
m o e th n Cm/ ne enme t s i a t m − o xp − 2m/ n fo rarg e ug(3 C
s
Th )ef o−r e t h e p b a b i l y h t uc
the
n be
three − hyphen u ni rm h y erg r p h on n] w h o e
hy ered ge s a rt he hyphen − threeerm
ari hmet c p r ore si no co nt ai−n ed n − i[n]One c 3)neas i ych ck t a l fo r ev
er
l ≥ 2 eh n umb( )o − f y cls−e o f l − en gth l i n G( i s at mo
t ( 3 ) . F r a2 su c hcyc ef G3 the p ro ba b i ty t at i t ap e a − r − s i n
F = F (R) s − ip
√
TP ge r − ef or etl eexp c t e d νm b er f − o c cl e in F sh o r t
er t − han g i at m os
R
se
y
wec
on
.
l = 23np )
= o( n)
Re frencs
[ A D R Y 94
N A on , R . A . D
k,
H Le f m an
R . Yu ster Thea lo ] r h i − ca p e
s−t
n − comma V . R ¨ d l
nd
mb n. 2(198
oc
F r a nkl,
P
1
(3 ,2 6−−
V R ¨o dl a nd R . W ilo
Then u
n
ix
ie n
6
mbe
o s ub m
su ls
t,
re
ric s o
typ e
a
H d a ma r m
t
a
d re te d
J . C om
T heo S r . B 44 (19 8)31 – 3 2 8
HF u r
ten be g,E rdo c behav o
odiag
Se m er´ di o n rt me t
pr ogrs io ns,J. A n a .Ma th . 1(19 7), 24−− 2
R
L.G
ra h
m
and
rthm e i
p
g e s i ns , . om binT
e or
Sr A 4 21986)2 70 – 6 7
R . LG r ah a m andV. R ¨dl N u m er s in R a
m sey
h
oy, in:
torc
rveys C
m bi a
s
187, C .
b
n
er123,
Cm
b dge U
n
v ry
i e it Pr
e s
C
r ige , 198, 111
amb
u
153
a
Y oh ay ak awa an
d
TL u
zak , he ind u e
sze − Ra
xel, K
. m
e
n
m ero cy l s
C
mbn . P
oa b C
put (19 9), 27 – 239
a thm e t
pro r es
D .R. Heat h− Br o w n
I
neg res s conta nng n
9
. odo nM a th S o
35 (1 8), 3
85 − − 4
S Jna so n, T . Lc aan
dA
Rui
ski , A exp o ent lbou nd frth p
o ba
iyo f n one xes n ce of as pe
f ied sbu gra h ina
ran
d om
A
ń
R
nom G a
s0 8, M K
a o ńs k J
Jawor k
a n
.
u
(
d R
iski eds
W iey 199 073 − −87 [ M D 8
C .H. Mc Diarm d,O nth e m
eho d
of b o ud
d n
s,
ed if f re c e i nS ur v
nC omb na o
c19 89J .Sie m
ons( ed .
,
Sr 14, C
mrdge U ni v r
yPs r s, C a mb dg e,1989 148−−1 8.
e
a
J . N eˇse r ˇ a nd
V . R ¨ o l , P a r it c o st ructi n
m ant h
rem
nd Ra se
m
i
fo rse ,ν mb s a
d
pC
m
nt. Ma th U n v
Crol
n. 2 (19 8
ces o
P . E .H
5 6 9 5 80
[ V 8
H
J
r¨ m e l an B
Vo gt As pr s
G raha m – o t schid t
eor m, rT an Ame .M a.th So c .30 ( 1988,11 3 – 1 3
V
Ro ¨ lO
n
Rms efy a mil ie of s e
, G ap hC
m
bi
6(1 990
ai
L
[o5
K
F. R o t h O n
r ns
eso in teer s, J .on do n Math.S o
28( 193
14 − −1
th
et r i an l es C o oM
q
IZR
ah
9
S oc.J´no sBol yi 18 (1 978)93945
JS
pn c
,R
trc
9( 17
[S z 7
rihm e
E
zem e
pr o re
s
´d , Onet
s
of
n
ed
Rm
se
yc o nfi
)2 78−−8 6
egers co
a
nnno
kee m
nts
na
aph
n − d Z . Palka ( eds . ) , An n D
screte M ath
Am
[T h 87 − −−,Rano m
g
aphs st r ny re g l ag r
andps eudo−r ndo mgrap hs,i
S uve y s in Cm bn a o c 198 7, C .
W hte ea ( e
. , L ndon
Mh
S
uLectre N teS e. 123 C a m rdgeU n i v er i
yPes Ca mb r d ge 1987 , 17
9.
M a em´t c a eE stat ś t c − i
ni−v e s d − i a
Sc
deS − tilde − a o Pau o
a
ndC omp uter
e
cn
o101 0 A d a m
Mck iew c − z U ni esit
M
a teki 4eight − slash4 f
M
o P alo, SP , Baz
l
A mtl − a n a − t, G e − orgia30322
U period − S. A .
R
eeive o n 9 . 6
( 280