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

Success Amplification RPsuppose A x

Krandombitsis a rand alg for L w

Tx cL Rgb Aix r YesHell Rgb Ahir yes s

gGoal error no more random bits

yesBy i t ii

qfyesif s b

lodchoose some S c S Is

LTls Sgd Ed Yg lod

so rls 413 E s Id