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❆♥② ❣r♦✉♣ ✇✐t❤ ✺ ♦r ♠♦r❡ ❡❧❡♠❡♥ts ❤❛s ❛ ♥♦♥✲♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤✳ ❆ ❣r♦✉♣ ♠❛② ❤❛✈❡ ♥♦♥✲✐s♦♠♦r♣❤✐❝ ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤s✳ ❇❛s✐❝ ❋❛❝ts ❆ ❣r❛♣❤ ✐s ♣❧❛♥❛r ✐✛ ✐t ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✐♥ t❤❡ ✷✲s♣❤❡r❡✱ S ✷ ❇❡✐♥❣ ♣❧❛♥❛r ✐s ❤❡r❡❞✐t❛r②✿ t❤❛t ✐s✱ s✉❜❣r♦✉♣s ♦❢ ♣❧❛♥❛r ❣r♦✉♣s ❛r❡ ♣❧❛♥❛r✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❆ ❣r♦✉♣ ♠❛② ❤❛✈❡ ♥♦♥✲✐s♦♠♦r♣❤✐❝ ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤s✳ ❇❛s✐❝ ❋❛❝ts ❆ ❣r❛♣❤ ✐s ♣❧❛♥❛r ✐✛ ✐t ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✐♥ t❤❡ ✷✲s♣❤❡r❡✱ S ✷ ❇❡✐♥❣ ♣❧❛♥❛r ✐s ❤❡r❡❞✐t❛r②✿ t❤❛t ✐s✱ s✉❜❣r♦✉♣s ♦❢ ♣❧❛♥❛r ❣r♦✉♣s ❛r❡ ♣❧❛♥❛r✳ ❆♥② ❣r♦✉♣ ✇✐t❤ ✺ ♦r ♠♦r❡ ❡❧❡♠❡♥ts ❤❛s ❛ ♥♦♥✲♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❇❛s✐❝ ❋❛❝ts ❆ ❣r❛♣❤ ✐s ♣❧❛♥❛r ✐✛ ✐t ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✐♥ t❤❡ ✷✲s♣❤❡r❡✱ S ✷ ❇❡✐♥❣ ♣❧❛♥❛r ✐s ❤❡r❡❞✐t❛r②✿ t❤❛t ✐s✱ s✉❜❣r♦✉♣s ♦❢ ♣❧❛♥❛r ❣r♦✉♣s ❛r❡ ♣❧❛♥❛r✳ ❆♥② ❣r♦✉♣ ✇✐t❤ ✺ ♦r ♠♦r❡ ❡❧❡♠❡♥ts ❤❛s ❛ ♥♦♥✲♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤✳ ❆ ❣r♦✉♣ ♠❛② ❤❛✈❡ ♥♦♥✲✐s♦♠♦r♣❤✐❝ ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤s✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

t❤❡r❡ ✐s ♥♦ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ ❞❡t❡r♠✐♥❡s✱ ❢♦r ❛♥② ❣✐✈❡♥ ♣r❡s❡♥t❛t✐♦♥✱ ✇❤❡t❤❡r ♦r ♥♦t t❤❡ ❣r♦✉♣ ❤❛s ❛ ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤✳ ❇❛s✐❝ ❋❛❝ts P♦ss❡ss✐♥❣ ❛ ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤ ✐s ❛ ▼❛r❦♦✈ Pr♦♣❡rt②✿ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❇❛s✐❝ ❋❛❝ts P♦ss❡ss✐♥❣ ❛ ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤ ✐s ❛ ▼❛r❦♦✈ Pr♦♣❡rt②✿ t❤❡r❡ ✐s ♥♦ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ ❞❡t❡r♠✐♥❡s✱ ❢♦r ❛♥② ❣✐✈❡♥ ♣r❡s❡♥t❛t✐♦♥✱ ✇❤❡t❤❡r ♦r ♥♦t t❤❡ ❣r♦✉♣ ❤❛s ❛ ♣❧❛♥❛r ❈❛②❧❡② ❣r❛♣❤✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✷ ✿ ✸ ✿ ■♥✜♥✐t❡ ❣r♦✉♣s ❊①❛♠♣❧❡s✿ · · · · · · Z ✿ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✸ ✿ ■♥✜♥✐t❡ ❣r♦✉♣s ❊①❛♠♣❧❡s✿ · · · · · · Z ✿ · · · · · · Z × Z ✷ ✿ · · · · · · P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

■♥✜♥✐t❡ ❣r♦✉♣s ❊①❛♠♣❧❡s✿ · · · · · · Z ✿ · · · · · · Z × Z ✷ ✿ · · · · · · · · · · · · · · · · · · · · · Z × Z ✸ ✿ · · · · · · · · · · · · P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

■♥✜♥✐t❡ ❣r♦✉♣s Z × Z ✸ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✇✐t❤ ✷ ❱❆Ps✿ ♦r ♦♥❡✿ ❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥ts ❆♥② ✐♥✜♥✐t❡ ❣r❛♣❤ ❡♠❜❡❞❞❡❞ ✐♥ S ✷ ✇✐❧❧ ❤❛✈❡ ❛ ❱❡rt❡① ❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥t✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

♦r ♦♥❡✿ ❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥ts ❆♥② ✐♥✜♥✐t❡ ❣r❛♣❤ ❡♠❜❡❞❞❡❞ ✐♥ S ✷ ✇✐❧❧ ❤❛✈❡ ❛ ❱❡rt❡① ❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥t✳ Z ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✇✐t❤ ✷ ❱❆Ps✿ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥ts ❆♥② ✐♥✜♥✐t❡ ❣r❛♣❤ ❡♠❜❡❞❞❡❞ ✐♥ S ✷ ✇✐❧❧ ❤❛✈❡ ❛ ❱❡rt❡① ❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥t✳ Z ❝❛♥ ❜❡ ❡♠❜❡❞❞❡❞ ✇✐t❤ ✷ ❱❆Ps✿ ♦r ♦♥❡✿ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥ts ❚❤❡ s❡t ♦❢ ❱❆Ps ✭ A ✮ ✐s ❡ss❡♥t✐❛❧ ✐❢ ❛♥② t✇♦ ♦❢ ✐ts ♣♦✐♥ts ❛r❡ s❡♣❛r❛t❡❞ ❜② ❛ ❝✐r❝✉✐t ♦❢ Γ ✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❊ss❡♥t✐❛❧ ❆❝❝✉♠✉❧❛t✐♦♥ P♦✐♥ts P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✸ ✷ ✸ ✶ ✹ ❊①❛♠♣❧❡ � � t , r | t ✸ = r ✷ = ( tr ) ✷ = ✶ S ✸ = P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✸ ✷ ✸ ✶ ✹ ❊①❛♠♣❧❡ � � t , r | t ✸ = r ✷ = ( tr ) ✷ = ✶ S ✸ = P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❊①❛♠♣❧❡ � � � � t , r | t ✸ = r ✷ = ( tr ) ✷ = ✶ t , g | t ✸ = g ✷ = ( tg ) ✸ = ✶ S ✸ = A ✹ = P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✸ ✷ ✷ ✷ ✸ ✶ ✸ ✹ ❊①❛♠♣❧❡ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✸ ✷ ✷ ✷ ✸ ✶ ✸ ✹ ❊①❛♠♣❧❡ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✸ ✷ ✷ ✷ ✸ ✶ ✸ ✹ ❊①❛♠♣❧❡ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✸ ✷ ✷ ✷ ✸ ✶ ✸ ✹ ❊①❛♠♣❧❡ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❊①❛♠♣❧❡ � � t , r , g | t ✸ = r ✷ = g ✷ = ( tr ) ✷ = ( tg ) ✸ = ✶ S ✸ ∗ � t � A ✹ = P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

■♥ ❢❛❝t✱ ✐s ♣❡r❢❡❝t ✭♥♦♥❡♠♣t②✱ ❝❧♦s❡❞✱ ✇✐t❤ ♥♦ ✐s♦❧❛t❡❞ ♣♦✐♥ts✮ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞✳ ▼♦r❡ t❤❛♥ ♦♥❡ ❱❆P ❚❤❡♦r❡♠ ❙✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| > ✷ ✳ ❚❤❡♥ A ✐s ✉♥❝♦✉♥t❛❜❧❡✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

▼♦r❡ t❤❛♥ ♦♥❡ ❱❆P ❚❤❡♦r❡♠ ❙✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| > ✷ ✳ ❚❤❡♥ A ✐s ✉♥❝♦✉♥t❛❜❧❡✳ ■♥ ❢❛❝t✱ A ✐s ♣❡r❢❡❝t ✭♥♦♥❡♠♣t②✱ ❝❧♦s❡❞✱ ✇✐t❤ ♥♦ ✐s♦❧❛t❡❞ ♣♦✐♥ts✮ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

▲♦✇ ❈♦♥♥❡❝t✐✈✐t② < a , b , t | a ✷ = b ✷ = ( ab ) ✷ = ✶ , t − ✶ at = b > P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

▲♦✇ ❈♦♥♥❡❝t✐✈✐t② < a , b , t | a ✷ = b ✷ = ( ab ) ✷ = ✶ , t − ✶ at = b > P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

▲♦✇ ❈♦♥♥❡❝t✐✈✐t② < a , b , t | a ✷ = b ✷ = ( ab ) ✷ = ✶ , t − ✶ at = b > P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

▲♦✇ ❈♦♥♥❡❝t✐✈✐t② < a , b , t | a ✷ = b ✷ = ( ab ) ✷ = ✶ , t − ✶ at = b > P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

■❢ ✵✱ ✐s ✜♥✐t❡ ■❢ ✶✱ ✐s ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ ♦❢ ✐s♦♠❡tr✐❡s ♦❢ ❡✐t❤❡r t❤❡ ❡✉❝❧✐❞❡❛♥ ♦r t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ✏✇❛❧❧♣❛♣❡r ❣r♦✉♣✳✑ ❲❡ r❡❢❡r t♦ t❤❡s❡ ❝♦❧❧❡❝t✐✈❡❧② ❛s str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s✳ ❙tr✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s ❤❛✈❡ ❜❡❡♥ ❝❛t❛❧♦❣✉❡❞✳ ❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ ❛♥❞ A ✐s ❡ss❡♥t✐❛❧✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

■❢ ✶✱ ✐s ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ ♦❢ ✐s♦♠❡tr✐❡s ♦❢ ❡✐t❤❡r t❤❡ ❡✉❝❧✐❞❡❛♥ ♦r t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ✏✇❛❧❧♣❛♣❡r ❣r♦✉♣✳✑ ❲❡ r❡❢❡r t♦ t❤❡s❡ ❝♦❧❧❡❝t✐✈❡❧② ❛s str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s✳ ❙tr✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s ❤❛✈❡ ❜❡❡♥ ❝❛t❛❧♦❣✉❡❞✳ ❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ ❛♥❞ A ✐s ❡ss❡♥t✐❛❧✳ ■❢ |A| = ✵✱ G ✐s ✜♥✐t❡ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❋♦r ❡①❛♠♣❧❡✱ ❛ ✏✇❛❧❧♣❛♣❡r ❣r♦✉♣✳✑ ❲❡ r❡❢❡r t♦ t❤❡s❡ ❝♦❧❧❡❝t✐✈❡❧② ❛s str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s✳ ❙tr✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s ❤❛✈❡ ❜❡❡♥ ❝❛t❛❧♦❣✉❡❞✳ ❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ ❛♥❞ A ✐s ❡ss❡♥t✐❛❧✳ ■❢ |A| = ✵✱ G ✐s ✜♥✐t❡ ■❢ |A| = ✶✱ G ✐s ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ ♦❢ ✐s♦♠❡tr✐❡s ♦❢ ❡✐t❤❡r t❤❡ ❡✉❝❧✐❞❡❛♥ ♦r t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❲❡ r❡❢❡r t♦ t❤❡s❡ ❝♦❧❧❡❝t✐✈❡❧② ❛s str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s✳ ❙tr✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s ❤❛✈❡ ❜❡❡♥ ❝❛t❛❧♦❣✉❡❞✳ ❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ ❛♥❞ A ✐s ❡ss❡♥t✐❛❧✳ ■❢ |A| = ✵✱ G ✐s ✜♥✐t❡ ■❢ |A| = ✶✱ G ✐s ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ ♦❢ ✐s♦♠❡tr✐❡s ♦❢ ❡✐t❤❡r t❤❡ ❡✉❝❧✐❞❡❛♥ ♦r t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ✏✇❛❧❧♣❛♣❡r ❣r♦✉♣✳✑ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❙tr✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s ❤❛✈❡ ❜❡❡♥ ❝❛t❛❧♦❣✉❡❞✳ ❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ ❛♥❞ A ✐s ❡ss❡♥t✐❛❧✳ ■❢ |A| = ✵✱ G ✐s ✜♥✐t❡ ■❢ |A| = ✶✱ G ✐s ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ ♦❢ ✐s♦♠❡tr✐❡s ♦❢ ❡✐t❤❡r t❤❡ ❡✉❝❧✐❞❡❛♥ ♦r t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ✏✇❛❧❧♣❛♣❡r ❣r♦✉♣✳✑ ❲❡ r❡❢❡r t♦ t❤❡s❡ ❝♦❧❧❡❝t✐✈❡❧② ❛s str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ ❛♥❞ A ✐s ❡ss❡♥t✐❛❧✳ ■❢ |A| = ✵✱ G ✐s ✜♥✐t❡ ■❢ |A| = ✶✱ G ✐s ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ ♦❢ ✐s♦♠❡tr✐❡s ♦❢ ❡✐t❤❡r t❤❡ ❡✉❝❧✐❞❡❛♥ ♦r t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ✏✇❛❧❧♣❛♣❡r ❣r♦✉♣✳✑ ❲❡ r❡❢❡r t♦ t❤❡s❡ ❝♦❧❧❡❝t✐✈❡❧② ❛s str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s✳ ❙tr✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣s ❤❛✈❡ ❜❡❡♥ ❝❛t❛❧♦❣✉❡❞✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✐s ❝❧♦s❡❞ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞ ✷ ✐s ♦♣❡♥ ❛♥❞ ❝♦♥♥❡❝t❡❞✳ ✷ ❤❛s ❛ ❝♦♥♥❡❝t❡❞ ✉♥✐✈❡rs❛❧ ❝♦✈❡r✐♥❣ s♣❛❝❡ ❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| ≥ ✷ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✷ ✐s ♦♣❡♥ ❛♥❞ ❝♦♥♥❡❝t❡❞✳ ✷ ❤❛s ❛ ❝♦♥♥❡❝t❡❞ ✉♥✐✈❡rs❛❧ ❝♦✈❡r✐♥❣ s♣❛❝❡ ❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| ≥ ✷ A ✐s ❝❧♦s❡❞ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✷ ❤❛s ❛ ❝♦♥♥❡❝t❡❞ ✉♥✐✈❡rs❛❧ ❝♦✈❡r✐♥❣ s♣❛❝❡ ❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| ≥ ✷ A ✐s ❝❧♦s❡❞ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞ ⇒ U = S ✷ − A ✐s ♦♣❡♥ ❛♥❞ ❝♦♥♥❡❝t❡❞✳ = P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| ≥ ✷ A ✐s ❝❧♦s❡❞ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞ ⇒ U = S ✷ − A ✐s ♦♣❡♥ ❛♥❞ ❝♦♥♥❡❝t❡❞✳ = ⇒ U ❤❛s ❛ ❝♦♥♥❡❝t❡❞ ✉♥✐✈❡rs❛❧ ❝♦✈❡r✐♥❣ s♣❛❝❡ � U ≈ R ✷ = P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| ≥ ✷ A ✐s ❝❧♦s❡❞ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞ ⇒ U = S ✷ − A ✐s ♦♣❡♥ ❛♥❞ ❝♦♥♥❡❝t❡❞✳ = ⇒ U ❤❛s ❛ ❝♦♥♥❡❝t❡❞ ✉♥✐✈❡rs❛❧ ❝♦✈❡r✐♥❣ s♣❛❝❡ � U ≈ R ✷ = � U U P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| ≥ ✷ A ✐s ❝❧♦s❡❞ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞ ⇒ U = S ✷ − A ✐s ♦♣❡♥ ❛♥❞ ❝♦♥♥❡❝t❡❞✳ = ⇒ U ❤❛s ❛ ❝♦♥♥❡❝t❡❞ ✉♥✐✈❡rs❛❧ ❝♦✈❡r✐♥❣ s♣❛❝❡ � U ≈ R ✷ = � � U U U U Γ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❲❡ s✉♣♣♦s❡ Γ ✐s ✸✲❝♦♥♥❡❝t❡❞✱ A ✐s ❡ss❡♥t✐❛❧✱ ❛♥❞ |A| ≥ ✷ A ✐s ❝❧♦s❡❞ ❛♥❞ t♦t❛❧❧② ❞✐s❝♦♥♥❡❝t❡❞ ⇒ U = S ✷ − A ✐s ♦♣❡♥ ❛♥❞ ❝♦♥♥❡❝t❡❞✳ = ⇒ U ❤❛s ❛ ❝♦♥♥❡❝t❡❞ ✉♥✐✈❡rs❛❧ ❝♦✈❡r✐♥❣ s♣❛❝❡ � U ≈ R ✷ = � � � � U U U Γ U U U Γ Γ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✐s ❡♠❜❡❞❞❡❞ ✐♥ ✇✐t❤ ♥♦ ❱❆Ps✱ s♦ ✐s ❛ str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣✳ ❚❤❡♦r❡♠ � Γ ✐s t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦❢ ❛ ❣r♦✉♣ � G ✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❚❤❡♦r❡♠ � Γ ✐s t❤❡ ❈❛②❧❡② ❣r❛♣❤ ♦❢ ❛ ❣r♦✉♣ � G ✳ � Γ ✐s ❡♠❜❡❞❞❡❞ ✐♥ � U ✇✐t❤ ♥♦ ❱❆Ps✱ s♦ � G ✐s ❛ str✐❝t❧② ♣❧❛♥❛r ❣r♦✉♣✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✷ ✷ ✶ ✶ ❊①❛♠♣❧❡ G = � r , b , t | · · · � P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✷ ✷ ✶ ✶ ❊①❛♠♣❧❡ G = � r , b , t | · · · � P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✷ ✷ ✶ ✶ ❊①❛♠♣❧❡ � G = � r , b , t | · · · � G = � r , b , t P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❊①❛♠♣❧❡ � G = � r , b , t | r ✷ , b ✷ , rt − ✶ rt , bt − ✶ bt � G = � r , b , t | · · · � P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✷ ✷ ✷ ✸ ❊①❛♠♣❧❡ G = � r , g , t | · · · � P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✷ ✷ ✷ ✸ ❊①❛♠♣❧❡ G = � r , g , t | · · · � P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✷ ✷ ✷ ✸ ❊①❛♠♣❧❡ � G = � r , g , t | · · · � G = � r , g , t P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❊①❛♠♣❧❡ � G = � r , g , t | r ✷ , g ✷ , ( rt ) ✷ , ( gt ) ✸ � G = � r , g , t | · · · � P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❚❤❡♦r❡♠ ✱ ✇❤❡r❡ ✐s t❤❡ s❡t ♦❢ ❧❛❜❡❧s ♦❢ ❜♦✉♥❞❛r✐❡s ♦❢ ❞✐s❦s ✐♥ ✷ ❚❤❡♦r❡♠ ✱ ✇❤❡r❡ ✐s ❛♥② s❡t ♦❢ ✇♦r❞s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ ❛♥② ♣❛t❤ ✇❤♦s❡ ❧❛❜❡❧ ✐s ✐♥ ✐s ❛ s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ ❛♥② t✇♦ ❱❆Ps ❛r❡ s❡♣❛r❛t❡❞ ❜② ❛ ❝✐r❝✉✐t ♦❢ ✇❤♦s❡ ❧❛❜❡❧ ❜❡❧♦♥❣s t♦ G = � X | · · · � Γ ⊆ S ✷ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❚❤❡♦r❡♠ ✱ ✇❤❡r❡ ✐s ❛♥② s❡t ♦❢ ✇♦r❞s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ ❛♥② ♣❛t❤ ✇❤♦s❡ ❧❛❜❡❧ ✐s ✐♥ ✐s ❛ s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ ❛♥② t✇♦ ❱❆Ps ❛r❡ s❡♣❛r❛t❡❞ ❜② ❛ ❝✐r❝✉✐t ♦❢ ✇❤♦s❡ ❧❛❜❡❧ ❜❡❧♦♥❣s t♦ G = � X | · · · � Γ ⊆ S ✷ ❚❤❡♦r❡♠ � G = � X | S � ✱ ✇❤❡r❡ S ✐s t❤❡ s❡t ♦❢ ❧❛❜❡❧s ♦❢ ❜♦✉♥❞❛r✐❡s ♦❢ ❞✐s❦s ✐♥ S ✷ − (Γ ∪ A ) P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✱ ✇❤❡r❡ ✐s ❛♥② s❡t ♦❢ ✇♦r❞s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ ❛♥② ♣❛t❤ ✇❤♦s❡ ❧❛❜❡❧ ✐s ✐♥ ✐s ❛ s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ ❛♥② t✇♦ ❱❆Ps ❛r❡ s❡♣❛r❛t❡❞ ❜② ❛ ❝✐r❝✉✐t ♦❢ ✇❤♦s❡ ❧❛❜❡❧ ❜❡❧♦♥❣s t♦ G = � X | · · · � Γ ⊆ S ✷ ❚❤❡♦r❡♠ � G = � X | S � ✱ ✇❤❡r❡ S ✐s t❤❡ s❡t ♦❢ ❧❛❜❡❧s ♦❢ ❜♦✉♥❞❛r✐❡s ♦❢ ❞✐s❦s ✐♥ S ✷ − (Γ ∪ A ) ❚❤❡♦r❡♠ G = � X | S ∪ L � P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❛♥② t✇♦ ❱❆Ps ❛r❡ s❡♣❛r❛t❡❞ ❜② ❛ ❝✐r❝✉✐t ♦❢ ✇❤♦s❡ ❧❛❜❡❧ ❜❡❧♦♥❣s t♦ G = � X | · · · � Γ ⊆ S ✷ ❚❤❡♦r❡♠ � G = � X | S � ✱ ✇❤❡r❡ S ✐s t❤❡ s❡t ♦❢ ❧❛❜❡❧s ♦❢ ❜♦✉♥❞❛r✐❡s ♦❢ ❞✐s❦s ✐♥ S ✷ − (Γ ∪ A ) ❚❤❡♦r❡♠ G = � X | S ∪ L � ✱ ✇❤❡r❡ L ✐s ❛♥② s❡t ♦❢ ✇♦r❞s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ ❛♥② ♣❛t❤ ✇❤♦s❡ ❧❛❜❡❧ ✐s ✐♥ L ✐s ❛ s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ Γ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

G = � X | · · · � Γ ⊆ S ✷ ❚❤❡♦r❡♠ � G = � X | S � ✱ ✇❤❡r❡ S ✐s t❤❡ s❡t ♦❢ ❧❛❜❡❧s ♦❢ ❜♦✉♥❞❛r✐❡s ♦❢ ❞✐s❦s ✐♥ S ✷ − (Γ ∪ A ) ❚❤❡♦r❡♠ G = � X | S ∪ L � ✱ ✇❤❡r❡ L ✐s ❛♥② s❡t ♦❢ ✇♦r❞s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ ❛♥② ♣❛t❤ ✇❤♦s❡ ❧❛❜❡❧ ✐s ✐♥ L ✐s ❛ s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ Γ ❛♥② t✇♦ ❱❆Ps ❛r❡ s❡♣❛r❛t❡❞ ❜② ❛ ❝✐r❝✉✐t ♦❢ Γ ✇❤♦s❡ ❧❛❜❡❧ ❜❡❧♦♥❣s t♦ L P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✸ ✷ ✷ ✶ ✶ ✸ ❊①❛♠♣❧❡✱ ❝♦♥t✐♥✉❡❞ � G = � r , b , t | r ✷ , b ✷ , rt − ✶ rt , bt − ✶ bt � P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✷ ✷ ✶ ✶ ✸ ❊①❛♠♣❧❡✱ ❝♦♥t✐♥✉❡❞ � G = � r , b , t | r ✷ , b ✷ , rt − ✶ rt , bt − ✶ bt � L = { ( rb ) ✸ } P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❊①❛♠♣❧❡✱ ❝♦♥t✐♥✉❡❞ � G = � r , b , t | r ✷ , b ✷ , rt − ✶ rt , bt − ✶ bt � L = { ( rb ) ✸ } G = � a , b , t | r ✷ , b ✷ , rt − ✶ rt , bt − ✶ bt , ( rb ) ✸ � P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✸ ✷ ✷ ✷ ✸ ✸ ❊①❛♠♣❧❡✱ ❝♦♥t✐♥✉❡❞ � G = � r , g , t | { r ✷ , g ✷ , ( rt ) ✷ , ( gt ) ✸ � P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✷ ✷ ✷ ✸ ✸ ❊①❛♠♣❧❡✱ ❝♦♥t✐♥✉❡❞ � G = � r , g , t | { r ✷ , g ✷ , ( rt ) ✷ , ( gt ) ✸ � L = { t ✸ } P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❊①❛♠♣❧❡✱ ❝♦♥t✐♥✉❡❞ � G = � r , g , t | { r ✷ , g ✷ , ( rt ) ✷ , ( gt ) ✸ � L = { t ✸ } G = � r , g , t | r ✷ , g ✷ , ( rt ) ✷ , ( gt ) ✸ , t ✸ � P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❋♦r ❛ ❣✐✈❡♥ ✱ ✇❤❛t ❛r❡ t❤❡ ♣♦ss✐❜✐❧✐t✐❡s ❢♦r t❤❡ s❡t ❄ ❚❤❡ ◗✉❡st✐♦♥✱ r❡❢♦r♠✉❧❛t❡❞ ❲❡ ❦♥♦✇ ✇❤❛t � G ❝❛♥ ❜❡✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❚❤❡ ◗✉❡st✐♦♥✱ r❡❢♦r♠✉❧❛t❡❞ ❲❡ ❦♥♦✇ ✇❤❛t � G ❝❛♥ ❜❡✳ ❋♦r ❛ ❣✐✈❡♥ � G ✱ ✇❤❛t ❛r❡ t❤❡ ♣♦ss✐❜✐❧✐t✐❡s ❢♦r t❤❡ s❡t L ❄ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❤❛s ❛ ♣r❡s❡♥t❛t✐♦♥ ❛s ❛❜♦✈❡✳ ✱ ✇❤❡r❡ ✐s t❤❡ ❧❛❜❡❧ ♦❢ ❛♥② s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ t❤❛t s❡♣❛r❛t❡s t❤❡ t✇♦ ❱❆Ps✳ ❆♣♣❧✐❝❛t✐♦♥✿ ❡♠❜❡❞❞✐♥❣s ✇✐t❤ ✷ ❱❆Ps ❙✉♣♣♦s❡ Γ ✐s ❡♠❜❡❞❞❡❞ ✐♥ S ✷ ✇✐t❤ ✷ ❱❆Ps✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

✱ ✇❤❡r❡ ✐s t❤❡ ❧❛❜❡❧ ♦❢ ❛♥② s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ t❤❛t s❡♣❛r❛t❡s t❤❡ t✇♦ ❱❆Ps✳ ❆♣♣❧✐❝❛t✐♦♥✿ ❡♠❜❡❞❞✐♥❣s ✇✐t❤ ✷ ❱❆Ps ❙✉♣♣♦s❡ Γ ✐s ❡♠❜❡❞❞❡❞ ✐♥ S ✷ ✇✐t❤ ✷ ❱❆Ps✳ � G ❤❛s ❛ ♣r❡s❡♥t❛t✐♦♥ � X | S � ❛s ❛❜♦✈❡✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❆♣♣❧✐❝❛t✐♦♥✿ ❡♠❜❡❞❞✐♥❣s ✇✐t❤ ✷ ❱❆Ps ❙✉♣♣♦s❡ Γ ✐s ❡♠❜❡❞❞❡❞ ✐♥ S ✷ ✇✐t❤ ✷ ❱❆Ps✳ � G ❤❛s ❛ ♣r❡s❡♥t❛t✐♦♥ � X | S � ❛s ❛❜♦✈❡✳ L = { ℓ } ✱ ✇❤❡r❡ ℓ ✐s t❤❡ ❧❛❜❡❧ ♦❢ ❛♥② s✐♠♣❧❡ ❝✐r❝✉✐t ✐♥ Γ t❤❛t s❡♣❛r❛t❡s t❤❡ t✇♦ ❱❆Ps✳ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❤❛s ❧❛❜❡❧ ❀ ❤❛s ❧❛❜❡❧ ✶ ✶ ✳ ✳ ✳ ❚❤❡ ♣❛t❤ ✐s ❤♦♠♦t♦♣✐❝ ❡✐t❤❡r t♦ ♦r t♦ v p Γ p ❤❛s ❧❛❜❡❧ ℓ ❀ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s

❤❛s ❧❛❜❡❧ ✶ ✶ ✳ ✳ ✳ ❚❤❡ ♣❛t❤ ✐s ❤♦♠♦t♦♣✐❝ ❡✐t❤❡r t♦ ♦r t♦ v ′ v v q p p Γ Γ p ❤❛s ❧❛❜❡❧ ℓ ❀ q ❤❛s ❧❛❜❡❧ w ❀ P❧❛♥❛r ❈❛②❧❡② ●r❛♣❤s