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❆ ❙✉❜❧✐♥❡❛r ❯♣♣❡r ❇♦✉♥❞ ♦♥ ❖✉r ❈♦♠❜✐♥❛t♦r✐❛❧ Pr♦❜❧❡♠ ❆❧❡①❛♥❞❡r ❘✳ ❇❧♦❝❦ ❖❝t♦❜❡r ✷✱ ✷✵✶✼ ✶ ✴ ✸✶

✶ ❖✉r ❈♦♠❜✐♥❛t♦r✐❛❧ Pr♦❜❧❡♠ ✷ ❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ✸ Pr♦♦❢ ♦❢ ❖✉r ❘❡s✉❧t ✹ ❈♦♥♥❡❝t✐♦♥ t♦ t❤❡ ❚r✐✲❈♦❧♦r❡❞ ❙✉♠✲❋r❡❡ ❙❡t Pr♦❜❧❡♠ ✷ ✴ ✸✶

❖✉r ❈♦♠❜✐♥❛t♦r✐❛❧ Pr♦❜❧❡♠ ■♥ ♦✉r ❈❘❨P❚❖ ✷✵✶✼ P❛♣❡r ❬❇▼◆✶✼❪✱ ✇❡ ✐♥tr♦❞✉❝❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠ ❖✉r Pr♦❜❧❡♠ ❋✐♥❞ t✇♦ ♦r❞❡r❡❞ s❡ts S = { s 1 , . . . , s m } ❛♥❞ T = { t 1 , . . . , t m } s✉❝❤ t❤❛t s i ❛♥❞ t i ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs ❢♦r ❡✈❡r② i s i + t j < n ❢♦r ❡✈❡r② i, j s i + t j � = s k + t k ❢♦r ❡✈❡r② i, j, k t❤❛t ❛r❡ ♥♦t ❛❧❧ ✐❞❡♥t✐❝❛❧ m ✐s ♠❛①✐♠✐③❡❞ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ✉♣♣❡r ❜♦✉♥❞✐♥❣ m ◗✉❡st✐♦♥ ❋♦r ❧❛r❣❡ ❡♥♦✉❣❤ n ✱ ✐s m = O ( n ) ♣♦ss✐❜❧❡❄ m = O ( n ) ✐s t❤❡ ❜❡st ❜♦✉♥❞ ✇❡ ❝❛♥ ❤♦♣❡ ❢♦r ✸ ✴ ✸✶

❖✉r ❘❡s✉❧ts ❲❡ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ❚❤❡♦r❡♠ ✭❖✉r Pr♦❜❧❡♠ ✐s ❙✉❜❧✐♥❡❛r✮ � n � ❋♦r ♦✉r ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦❜❧❡♠✱ m = O ✳ log ∗ n ❍❡r❡ log ∗ ✐s ❞❡✜♥❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ❜❛s❡ ✷ ❛s � 0 n � 1 log ∗ n = 1 + log ∗ (log n ) n > 1 ❚♦ ❣❛✐♥ ✐♥s✐❣❤t ♦♥ ❤♦✇ t♦ ♣r♦✈❡ t❤✐s r❡s✉❧t✱ ✇❡ ❡①❛♠✐♥❡ s✐♠✐❧❛r ♣r♦❜❧❡♠s ❛♥❞ t❤❡ t❡❝❤♥✐q✉❡s ✉s❡❞ t♦ ♣r♦✈❡ t❤❡♠ ✹ ✴ ✸✶

❈♦♥♥❡❝t✐♦♥✿ ✸✲❋r❡❡ ❙❡ts ❖✉r ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦❜❧❡♠ ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ✇❡❧❧✲❦♥♦✇ ✸✲❢r❡❡ s❡t ♣r♦❜❧❡♠ ✸✲❋r❡❡ ❙❡t Pr♦❜❧❡♠ ❋✐♥❞ A ⊆ { 1 , . . . , n } s✉❝❤ t❤❛t A ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② ✸✲t❡r♠ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ | A | ✐s ♠❛①✐♠✐③❡❞ ❋♦r ♦✉r ♣r♦❜❧❡♠✱ s❡tt✐♥❣ S = T ②✐❡❧❞s t❤❡ ✸✲❋r❡❡ ❙❡t Pr♦❜❧❡♠ ❯♣♣❡r ❜♦✉♥❞s ❢♦r t❤❡ ✸✲❋r❡❡ ❙❡t ♣r♦❜❧❡♠ ♦r✐❣✐♥❛t❡❞ ✇✐t❤ ❘♦t❤ ❬❘♦t✺✸❪ ❘♦t❤✬s ❚❤❡♦r❡♠ s❤♦✇s t❤❛t | A | = o ( n ) ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❘♦t❤ ✉s❡s ❋♦✉r✐❡r ❛♥❛❧②s✐s t♦ s❤♦✇ | A | = O ( n/ (log log n )) ❇❡st ❦♥♦✇♥ ✉♣♣❡r ❜♦✉♥❞✿ O ( n (log log n ) 4 / log n ) ❛♥❞ ✇❛s s❤♦✇♥ ❜② ❇❧♦♦♠ ❬❇❧♦✶✻❪✱ ✉s✐♥❣ ❤❡❛✈② ♠❛❝❤✐♥❡r② ❢r♦♠ ❛❞❞✐t✐✈❡ ❝♦♠❜✐♥❛t♦r✐❝s ✺ ✴ ✸✶

Pr♦♦❢ ♦❢ ❛ ❙✉❜❧✐♥❡❛r ❯♣♣❡r ❇♦✉♥❞ ❖✉r ❣♦❛❧ ✐s t♦ s❤♦✇ s✉❜❧✐♥❡❛r✐t② ♦❢ ❖✉r ❈♦♠❜✐♥❛t♦r✐❛❧ Pr♦❜❧❡♠ ❚❤❡ ♠❛❝❤✐♥❡r② ✐♥ ♠❛♥② ♦❢ t❤❡ ♣r♦♦❢s ❢♦r ✸✲❋r❡❡ ❙❡ts ❞♦ ♥♦t ❡❛s✐❧② ❣❡♥❡r❛❧✐③❡ t♦ ❛♥❛❧②s✐s ♦✈❡r ✷ s❡ts ◗✉❡st✐♦♥ ■s t❤❡r❡ ❛ ♣r♦♦❢ ♦❢ s✉❜❧✐♥❡❛r✐t② ♦❢ ✸✲❋r❡❡ ❙❡ts t❤❛t ❝❛♥ ❣❡♥❡r❛❧✐③❡ t♦ ♦✉r ♣r♦❜❧❡♠❄ ❆♥s✇❡r ❨❡s✦ ❲❡ ❝❛♥ ✉s❡ ❛ ●r❛♣❤ ❚❤❡♦r❡t✐❝ r❡❞✉❝t✐♦♥ ❛♥❞ ✉s❡ ❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ✻ ✴ ✸✶

❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ❲❡✬❧❧ ❝❛r❡❢✉❧❧② ❞❡✜♥❡ ❛❧❧ t❤❡ ♥❡❝❡ss❛r② ❝♦♠♣♦♥❡♥ts ♥❡❡❞❡❞ ❢♦r ❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛✳ ▲❡t G = ( V, E ) ❜❡ ❛ s✐♠♣❧❡✱ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤ ♦♥ n ✈❡rt✐❝❡s ■❢ X ⊆ V ❛♥❞ Y ⊆ V ✱ ✇❡ ❞❡✜♥❡ e ( X, Y ) ❛s t❤❡ ♥✉♠❜❡r ♦❢ ❡❞❣❡s ❜❡t✇❡❡♥ X ❛♥❞ Y ❉❡✜♥✐t✐♦♥ ✭❊❞❣❡ ❉❡♥s✐t②✮ ❋♦r ❞✐s❥♦✐♥t X, Y ⊆ V ✱ ✇❡ ❞❡✜♥❡ t❤❡ ❡❞❣❡ ❞❡♥s✐t② ❛s d ( X, Y ) = e ( X, Y ) | X | · | Y | ✼ ✴ ✸✶

❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ◆❡①t ✇❡ ❤❛✈❡ t❤❡ ❘❡❣✉❧❛r✐t② ❈♦♥❞✐t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭ ε ✲❘❡❣✉❧❛r P❛✐rs✮ ▲❡t ε > 0 ✳ ●✐✈❡♥ ❞✐s❥♦✐♥t X, Y ⊆ V ✱ t❤❡ ♣❛✐r ( X, Y ) ✐s ε ✲r❡❣✉❧❛r ✐❢ ❢♦r ❡✈❡r② A ⊆ X ❛♥❞ B ⊆ Y s✉❝❤ t❤❛t | A | > ε | X | ❛♥❞ | B | > ε | Y | ✇❡ ❤❛✈❡ | d ( A, B ) − d ( X, Y ) | < ε ✽ ✴ ✸✶

❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ❚❤❡♦r❡♠ ✭❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ❬❙③❡✼✺❜✱ ❑❙❙❙✵✷❪✮ ∀ ε > 0 • ∃ M = M ( ε ) ∈ Z s✉❝❤ t❤❛t • ∀ s✐♠♣❧❡ ✉♥❞✐r❡❝t❡❞ G = ( V, E ) ♦♥ n ✈❡rt✐❝❡s • ∃ ❛ ♣❛rt✐t✐♦♥ ♦❢ V ✐♥t♦ k ❝❧❛ss❡s V 1 , V 2 , . . . , V k s✉❝❤ t❤❛t • k � M • | V i | � ⌈ εn ⌉ ❢♦r ❡✈❡r② i • || V i | − | V j || � 1 ❢♦r ❛❧❧ i, j ✭❡q✉✐♣❛rt✐t✐♦♥✮ • ( V i , V j ) ✐s ε ✲r❡❣✉❧❛r ✐♥ G ❢♦r ❛❧❧ ❡①❝❡♣t ❛t ♠♦st εk 2 ♣❛✐rs ( i, j ) ✾ ✴ ✸✶

❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ❚❤❡ ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ✐s ❛ ✈❡r② ♣♦✇❡r❢✉❧ t♦♦❧ ✐♥ ❊①tr❡♠❛❧ ●r❛♣❤ ❚❤❡♦r② ■♥t✉✐t✐✈❡ ❙t❛t❡♠❡♥t ❆♥② ❧❛r❣❡ ❛♥❞ ❞❡♥s❡ ❡♥♦✉❣❤ ❣r❛♣❤ ✏❧♦♦❦s ❧✐❦❡✑ ❛ ✉♥✐♦♥ ♦❢ ❛ s♠❛❧❧ ❛♠♦✉♥t ♦❢ r❛♥❞♦♠✲❧♦♦❦✐♥❣ ❜✐♣❛rt✐t❡ ❣r❛♣❤s✳ ▲❛r❣❡ ❡♥♦✉❣❤ ❣r❛♣❤s ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ❜② ❛ s♠❛❧❧ ♥✉♠❜❡r ♦❢ ✉♥✐❢♦r♠ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ❋♦r t❤✐s r❡❛s♦♥✱ t❤❡ ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ✐s ❛❧s♦ r❡❢❡rr❡❞ t♦ ❛s t❤❡ ❯♥✐❢♦r♠✐t② ▲❡♠♠❛ ◆♦t❡ ●♦✇❡rs ❬●♦✇✾✼❪ s❤♦✇❡❞ t❤❛t t❤❡ ❜♦✉♥❞ ♦❜t❛✐♥❡❞ ❢♦r M ( ε ) ✐s ❛ t♦✇❡r ♦❢ ✷✬s ♦❢ ❤❡✐❣❤t ♣r♦♣♦rt✐♦♥❛❧ t♦ ε − 5 ✳ ❍❡ s❤♦✇❡❞ t❤✐s ✐s ❛♥ ✐♥❤❡r❡♥t ❢❡❛t✉r❡ ♦❢ t❤❡ ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ✶✵ ✴ ✸✶

❙③❡♠❡ré❞✐✬s ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ ❙③❡♠❡ré❞✐ ❝r❡❛t❡❞ t❤✐s ❧❡♠♠❛ t♦ ♣r♦✈❡ ❤✐s ❢❛♠♦✉s r❡s✉❧t ❚❤❡♦r❡♠ ✭❙③❡♠❡ré❞✐✬s ❚❤❡♦r❡♠ ❬❙③❡✼✺❛❪✮ ∀ k � 3 ❛♥❞ δ > 0 • ∃ n 0 ∈ Z s✉❝❤ t❤❛t • ∀ N � n 0 • ■❢ A ⊆ { 1 , 2 , . . . , N } ❛♥❞ | A | � δN • t❤❡♥ A ❝♦♥t❛✐♥s ❛♥ ❛r✐t❤♠❡t✐❝ ♣r♦❣r❡ss✐♦♥ ♦❢ ❧❡♥❣t❤ k ❋♦r k = 3 ✱ ✇❡ ❤❛✈❡ ❡①❛❝t❧② ❘♦t❤✬s ❚❤❡♦r❡♠ ❢♦r ✸✲❋r❡❡ ❙❡ts ❲❡ ❢♦❧❧♦✇ t❤❡ ♣r♦♦❢ ♦❢ ❘♦t❤✬s ❚❤❡♦r❡♠ ✉s✐♥❣ t❤❡ ❘❡❣✉❧❛r✐t② ▲❡♠♠❛ t♦ ♣r♦✈❡ ♦✉r r❡s✉❧t ✶✶ ✴ ✸✶