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A. Juriˇ<strong>si</strong>ć in V. Batagelj: Verjetnostni račun in statistika 709 ✬ Erdöseva probabilistična metoda (1947) 34 točk določa 561 premic. Da se to zgodi v eni barvi, je verjetnost ✫ 2 −561 ≈ 2, 6 · 10 −169 . Velja tudi 1.000.000 165 34 = 3, 4 · 10 . Torej lahko pričakujemo 6 10 = 3, 4 · 10 34 165 ≈ 0, 01 oziroma 0, 01% enobarvnih. To pomeni, da v 99, 9% ne dobimo enobarvnega K34. Slednjo idejo pretvorimo v Erdösev dokaz.... Univerza v Ljubljani ▲ ❙ ▲ ▲ ● ❙ ▲ ▲ ☛ ✖ ▲ ✩ ✪
A. Juriˇ<strong>si</strong>ć in V. Batagelj: Verjetnostni račun in statistika 710 ✬ Dokaz Erdösevega izreka: probabilistična metoda (ni konstruktivna) in ˇstetje Naj bo Gn mnoˇzica grafov z vozliˇsči v1, v2, . . . , vn. Naj bo G k n mnoˇzica grafov iz Gn, ki vsebujejo k-kliko. Potem je |Gn| = 2 (n2) k , |Gn| = 2 (n2)−( k 2) n in Če je n < 2 k/2 , velja q ≤ q = |G k n|/|Gn| ≤ nk k 2 −( 2) . k! k2 2 2 −( k 2) k! Iz Gn = {G | G ∈ Gn} pa sledi, da manj kot polovica grafov iz Gn vsebuje k-antiklike. ✫ k < 1 2 . Se pravi, da manj kot polovica grafov iz Gn vsebuje k-klike. Univerza v Ljubljani ▲ ❙ ▲ ▲ ● ❙ ▲ ▲ ☛ ✖ ▲ ✩ ✪
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✬ ✫ Ljubljana, 6. oktober 2008
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A. Juriˇsić in V. Batagelj: Verje
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