Existenceofmagie alGraphs Lemma Tho Two no d 32 m Th J Chim d graph sit e t scL Hs Fd INDI Isd 1st FRI uisanbrofsomeves 2 V SCLJ dslsla.bz pCs s1s1 Yaeatwi p if atrandomandschow s T not a magical graph e l oo µ randomly o reV select ueR d times Ind R Rou that 12 holds whp L V sclj aalslsh z.VN lslXs T indicates Pfs CT if EXST so z holds p.gl sols E mY 5 g tags'd why A disconnecting fewer teens means low rank Times cays d simplifying meh n y2s g Sd assumption m pups insipid pigs o s d 2 Msps yd ndt jpsjd.rs
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mYNon explicit Existence of Magical Graphs I e Ii ii i z0V scL od1slsh INDI s 151 gfi d Proof sketch probmethod 0 Every vet chooses UER dtimesunit ind prove Pfob issatisfied L R Fix
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ExistenceofmagiealGraphs
LemmaThoTwo no d 32 m Th J Chimd graphsite t scL Hs Fd INDI Isd 1st
FRI uisanbrofsomeves2 VSCLJdslsla.bz pCs s1s1
Yaeatwip if atrandomandschow s
T
nota magicalgraph e l oo
µ randomly oreV select ueR d times IndRRou that 12 holds whp L
V sclj aalslsh z.VN lslXs T indicates Pfs CTif EXST so z holds
p.gl sols EmY5 g tags'd why
A disconnectingfewerteens
meanslowrank
Timescaysd
simplifying meh n y2s gSd
assumption
mpups insipid pigs
o
s d 2 Msps ydndt
jpsjd.rs
Non explicit Existence of Magical Graphs
I e
Ii i i ifiz0V scL od1slsh INDI s 151 g d
Proof sketch probmethod
0Every vet chooses UER d times unit ind
prove Pfobis satisfied L R
Fix T SCL Eaa Hs E ITI_Isl Xs indicates MOSTobserve if EXsT so then z is satisfied
S T
soTo Pg Xs to ESd got e
Prob xs.t tolss.EPnoblxsit.to sI th lFll nlsd
lhEPcyePcsagsd
neg.mg gd JSb
iss 5 Y Isd Pretend him
ftp ds
issued k 1by increasing d
Using magical graphs super concentrators
O 0Recursive construction
hen output complete bip graph
G un Sup cen n sup uh 4h
missA f use propertyL Rm n
1Hal's thinn Mgm Ks's
prove that Cw is a sup uh usey matchFix 5 T size k
SparsityE s v D t vei
setup
ErrorcorrectingcodesmagicalGraph
C s on tin subspace FzR
oi R
44omod2
dims 3 E rrate 3 Idistance un w c
w w 1W is a non zero c Lwtfw distCw wz id
Goal twee fo cutlets or hAssume for contradiction 1st ylad
ssfvlwcrl.to
T s 1st Ed S has a unique neighotherwise Hr
trials 4.151he unique nbr is a violated constraint