Probabilistic - P ro b a b ilis tic Me th o d Lecture 22...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 22: April 18 Pro b a b ilis tic   Me tho d
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Why Ra ndo m ne s s ? Probabilistic method : Pro ving  the  e xis te nc e  o f a n o b je c t s atis fying   c e rta in pro pe rtie s  witho ut ac tua lly c o ns truc ting  it. S ho w tha t a rando m  o b je c t s atis fying  tho s e   pro pe rtie s  with po s itive  pro b ab ility! That is , “c re a te ” the  de s ire d o b je c t b y flipping  c o ins !
Background image of page 2
Ra m s e y Num b e r G ive n a c o m ple te  g ra ph, we  want to  c o lo ur its  e dg e s  b y 2 c o lo urs . Ramsey number  R(k,l): is  the  s m alle s t num b e r n s o  tha t: Fo r e ve ry c o m ple te  g ra ph G  with a t le as t n ve rtic e s , the n any 2-c o lo uring  o f G  wo uld e ithe r has  a re d c o m ple te s ubg raph o f s ize  k (a  re d K(k)) o r a  b lue  c o m ple te  s ubg raph  o f s ize  l (a b lue  K(l)). R(k,l) Num b e r o f ve rtic e s The re  is  a c o lo uring  with  no  re d K(k) and no  b lue  K(l). Eve ry c o lo uring  m us t have a  re d K(k) o r a  b lue  K(l).
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Ra m s e y Num b e r R(3,3) R(3,3) Num b e r o f ve rtic e s The re  is  a  c o lo uring  with  no  re d K(3) a nd no  blue  K(3). Eve ry c o lo uring  m us t have a  re d K(3) o r a blue  K(3). Wha t is  R(3,3)? S o  R(3,3) > 5.
Background image of page 4
Ra m s e y Num b e r R(3,3) R(3,3) Num b e r o f ve rtic e s The re  is  a  c o lo uring  with  no  re d K(3) a nd no  blue  K(3). Eve ry c o lo uring  m us t have a  re d K(3) o r a blue  K(3). Claim : R(3,3)=6. Proof:  S uppo s e  the  e dg e s  o f a  c o m ple te  g raph o n 6 ve rtic e s  a re   c o lo ure d re d and blue . Pic k a  ve rte x v. The re  a re  5 e dg e s  inc ide nt to  v  and s o  a t le as t 3 o f the m  m us t b e  the  s am e  c o lo ur, s a y b lue . If any o f the  
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 19

Probabilistic - P ro b a b ilis tic Me th o d Lecture 22...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online