R 以 The probabilistic Method Paul Erdos
The Probabilistic Method Paul Erdős
Theorem (Erdos 1947) If(份-21-均<1 then it is possible to color the edges of Kn with two colors so that there is no monochromatic Kk subgraph. For each edge ee Kn, with prob 1/2 e is colored l0● with prob 1/2 For a particular Kk,(edges Pr[Kk Or K]=2l-匀
If n k ⇥ · 21 k 2 ⇥ < 1 then it is possible to color the edges of Kn with two colors so that there is no monochromatic Kk subgraph. Theorem (Erdős 1947) e is colored with prob 1/2 with prob 1/2 For a particular Kk , k 2 ⇥ edges Pr[ or ] = 21 k 2 ⇥ Kk Kk For each edge e Kn
Theorem (Erdos 1947) If(份-21-均<1 then it is possible to color the edges of Kn with two colors so that there is no monochromatic Kk subgraph. For a particular Kk, Pr[the Kk is monochromatic]=21-() number of Kk in Kn: () Pr[3a monochromatic Kk] ≤(2- ）<1
If n k ⇥ · 21 k 2 ⇥ < 1 then it is possible to color the edges of Kn with two colors so that there is no monochromatic Kk subgraph. Theorem (Erdős 1947) For a particular Kk , Pr[the Kk is monochromatic] = 21 k 2 ⇥ number of Kk in Kn: n k ⇥ Pr[ a monochromatic Kk ] ⇤ n k ⇥ · 21 k 2 ⇥ < 1
Theorem (Erdos 1947) If(份-21-均0 There exists a two-coloring without monochromatic Kk
If n k ⇥ · 21 k 2 ⇥ 0 There exists a two-coloring without monochromatic . Kk
Tournament T(V,E) n players,each pair has a match. u points to v iff u beats v. k-paradoxical: For every k-subset S of V, there is a player in S who 4 beats all players in S. "Does there exist a k-paradoxical tournament for every finite k?
Tournament n players, each pair has a match. u points to v iff u beats v. k-paradoxical: For every k-subset S of V, there is a player in V \ S who beats all players in S. T(V, E) “Does there exist a k-paradoxical tournament for every finite k?
Theorem(Erd6s 1963) If ()(1-)<1 then there is a k-paradoxical tournament of n players. Pick a random tournament T on n players [n]. Fixed any S∈ . Event As:no player in V\beat all players in S. PrAs=(1-2k)”-
If n k ⇥ 1 2k⇥nk < 1 then there is a k-paradoxical tournament of n players. Theorem (Erdős 1963) Pick a random tournament T on n players [n]. Fixed any S [n] k ⇥ Event AS : no player in V \S beat all players in S. Pr[AS] = 1 2k⇥nk
Theorem(Erd6s 1963) If()(1-2-k)”- 1 then there is a k-paradoxical tournament of n players. Pick a random tournament T on n players [n]. Event As:no player in \S beat all players in S. Pr[As=(1-2k)”-k Pr ≤∑(1-2k)-k <1 s∈() S∈()
If n k ⇥ 1 2k⇥nk < 1 then there is a k-paradoxical tournament of n players. Theorem (Erdős 1963) Pick a random tournament T on n players [n]. Event AS : no player in V \S beat all players in S. Pr[AS] = 1 2k⇥nk Pr < 1 ⇧ ⇤ ⌥ S( [n] k ) AS ⇥ ⌃ ⌅ ⇥ S⇥( [n] k ) (1 2k) nk
Theorem(Erd6s 1963) If ()(1-2-k)"<1 thon there is a k-paradoxical tournament of n players. Pick a random tournament T on n players [n]. Event As no player in \S beat all players in S. V<1 se() 0 S∈()
If n k ⇥ 1 2k⇥nk 0
Theorem(Erdos 1963) If (R)(1-2-k)"0 There is a k-paradoxical tournament on n players
If n k ⇥ 1 2k⇥nk 0 There is a k-paradoxical tournament on n players
The probabilistic Method Pick random ball from a box, 00 Pr[the ball is blue]>0 000 →There is a blue ball. Define a probability space O,and a property P: Pr[P(x)]>0 X >xE with the property P
The Probabilistic Method • Pick random ball from a box, Pr[the ball is blue]>0. ⇒ There is a blue ball. • Define a probability space Ω, and a property P: Pr x [P(x)] > 0 = ⇤x ⇥ with the property P