S is a clique, otherwise. of k-cliques: X=∑ Xs se) Goal: p=(n2-） Pr[X≥1]=o(1) p-“(）> Pr[X≥1]=1-o(1) Pr[X=0]=o(1)
XS = 1 S is a clique, 0 otherwise. # of k-cliques: Pr[X 1] = o(1) Pr[X ⇥ 1] = 1 o(1) Goal: 8S 2 ✓[n] k ◆ X = X S2( [n] k ) XS Pr[X = 0] = o(1) p = o ⇣ n2/(k1)⌘ p = ! ⇣ n2/(k1)⌘
sc(）-6 X=】 Xs otherwise. s∈() Goak卫=o(2-"）> Pr[X≥1]=o(1) x之-9e(货）t= ≤>Pr[is a clique] s∈()
XS = 1 S is a clique, 0 otherwise. Goal: Pr[X 1] = o(1) 8S 2 ✓[n] k ◆ X = X S2( [n] k ) XS p = o ⇣ n2/(k1)⌘ Pr[X 1] = Pr 9S 2 ✓[n] k ◆ s.t. XS = 1 X S2( [n] k ) Pr [S is a clique ]
s(太-8 S is a clique, X=】 otherwise. ∑Xs S∈() Gos:(e）今 P[X≥1]=o(1) Pe)ut ≤∑p() S∈() -o(n）-o四
XS = 1 S is a clique, 0 otherwise. Goal: Pr[X 1] = o(1) 8S 2 ✓[n] k ◆ X = X S2( [n] k ) XS p = o ⇣ n2/(k1)⌘ Pr[X 1] = Pr 9S 2 ✓[n] k ◆ s.t. XS = 1 X S2( [n] k ) p( k 2) = o ⇣ nk · n2( k 2)/(k1)⌘ = o (1)
X= otherwise. ∑Xs s∈() Goalk:p(n-2/)Prix =01-(1) PX=O≤PrX-EX]I≥EX]≤ Var X EX2 wa吗 S≠T =∑(E[X]-EXs2) s∈() ≤ ∑EX=E[X] s∈()
XS = 1 S is a clique, 0 otherwise. Goal: 8S 2 ✓[n] k ◆ X = X S2( [n] k ) XS p = ! Pr[X = 0] = o(1) ⇣ n2/(k1)⌘ Pr[X = 0] Pr[ |X E[X]| E[X] ] = X S2( [n] k ) Var [XS] + X S6=T Var[X] = Var Cov(XS, XT ) 2 6 4 X S2( [n] k ) XS 3 7 5 Var[X] E[X] 2 = X S2( [n] k ) (E⇥ X2 S ⇤ E [XS] 2 ) =E[X] X S2( [n] k ) E [XS]
se(8)太-68 X=∑Xs otherwise. s∈() Goal:p=(n=o(1) Pr[X=O≤PrIX-EX]I≥EX]I≤ Var X E[X]2 arx=r∑内 Var[Xsl+∑Cov(Xs,Xr) se()」se() S≠T ≤E[XW]+>Cov(Xs,Xr) S≠T
XS = 1 S is a clique, 0 otherwise. Goal: 8S 2 ✓[n] k ◆ X = X S2( [n] k ) XS p = ! Pr[X = 0] = o(1) ⇣ n2/(k1)⌘ Pr[X = 0] Pr[ |X E[X]| E[X] ] = X S2( [n] k ) Var [XS] + X S6=T Var[X] = Var Cov(XS, XT ) 2 6 4 X S2( [n] k ) XS 3 7 5 Var[X] E[X] 2 E [X] + X S6=T Cov(XS, XT )