- pm characteristic of field 0 / O w H ( a t b Y ' = at t b t - - PDF document

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- pm characteristic of field 0 / O w H ( a t b Y ' = at t b t - - PDF document

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1%20 FIELD I t I t . t 1 = O . . - pm characteristic of field 0 / O w H ( a t b Y ' = at t b t Hypergraph . , E ) vertex ( V set bertie laniary V E PCV ) E " IV Hh edges v IE km STERNER 'S TAM .m4I Uniform hypergraph : Hedge

slide-1
SLIDE 1

FIELD

1%20

I t I t

. . . t 1 = O
  • pmcharacteristic
  • f field

0/

w

O

H

( a t b Y

'

=att bt

slide-2
SLIDE 2

Hypergraph

.

(V

, E) vertex

V

set bertie laniary

E

E PCV)

"

edges

IVHh

v IE km

slide-3
SLIDE 3

STERNER'S TAM

.÷m4I

Uniform hypergraph :

Hedge

same size

km :@HEE)HAKk)

  • 2- uniform hyper

.oh

: GRAPH

slide-4
SLIDE 4

Bipartite graph

Oi

:#

ran

.

  • H
  • (V
, E)

"

¥.

incidence graph of Tl

slide-5
SLIDE 5

se

  • (v, E)

E -

  • { Ae ,
. . . , Am }

K EV

DEI

k

is

an independent set

it Kill Ai Ek)

slide-6
SLIDE 6

SET

CARDS

81=3

"

cards

  • V =ffz4

{ ={

"

SETS

" }

40,21471

T

T

affine lines

vertex

3

  • uniform

STS

Steiner triple system

  • ff
  • n table,

12 Cards

try to pick

a

SET

16 cards w/o

SET :

{ o, 134

( O ,

I , I

. I )
slide-7
SLIDE 7

DEF

INDEFENCE

NUMBER :

L(Jl) = wax size of an

indep

set

  • L ( SET gone) 216

look

up d( SETgear

)

  • d- dine SET:

STS

✓ = Ed

E : affine

  • lines

{Etbtltet) ←

b.to

slide-8
SLIDE 8

24216W Ld =L ( d -din EET)

24 E

xd

E 3d

vertices : (x

, . - xd)

d

Xi Eff

2 E

Fd

E3

the L

: - ein Id

'd

exists

2

E L E 3

Actually

↳2

was

TAM

L <3

major open

=

problem

slide-9
SLIDE 9

L exists

?

EI

fee Z di de

Super multiplicative

FEKETE's

Lenne

an sequence ,

an

> 0

,

seep meet .

then

F lima!

= sup a'

in

h→no

h

  • S

L

Z a'

¥

desk

toy

a

n

  • za

43

  • c)

d

L >2

  • c >% art
slide-10
SLIDE 10

HW

find int

. many

STS s

s.t .

Ltte) >I

#

Covering

number T

(transversal #)

'tan

Hitting number

hitting set

Vertex cover

← min

slide-11
SLIDE 11

Ded of

vertex : deg (a)

#il

ne Ai }

←edges

  • GREEDY

COVER ALGOR :

pick

vertex of highest degree

remove all incident edges

repeat

  • THMCL-ovisetgreedyaEE.am

E = win size of cover

N P- hard

even for graphs

slide-12
SLIDE 12

K -unit hypergraph

who

repeated

edges

⇒

me

  • Complete

k -uniform hypergraph:

m

k¥2

h m

= he (E) d k .

luhcnaknkraak

(H) = me x{ I Al l AEE}

slide-13
SLIDE 13

Chromatic

number

X

Ichi

woe air

.

c :V →{colors)

sit

.

no edge is

monochromatic

  • i. e. HAE E)

(I 4All 22)

X (Jl) = win # colors

in

legal coloring

slide-14
SLIDE 14
  • Je

k

  • uniform

k¥4

'

÷

.

we:④

:

Complete k -arif hyp

.

Knc

"

vertices

at

.¥I"÷*

I Lh

slide-15
SLIDE 15

"¥tn¥:*

↳

used

2 t

colors

KI

= E. n

  • K =3

case

n 75 ⇒X> 2

  • • PHP

general case HW

slide-16
SLIDE 16

m Cr )

= aim # edges of

an

r - unit . hypergraph

that

is

Not

2- adorable

  • BONUS

m Cr)

> I

  • I
  • ERDE

's

i.e

. if

ye r- unit

r
  • I

m

E 2

→

2 . adorable

  • random coloring

shorP(bga

CH

m(r ) e

  • c. r2
.I

ERDO'S

  • ie

m

slide-17
SLIDE 17

matching :

set of disjoint edges

O O O

z(Jl)

=wax # dig edges

( hee

  • ①

so E t

②

c- Ek

. so if R

K- unit

  • "dual quantities

"

systems

  • f
  • Linear programmingFinis