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❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✶ ❚♦❞❛②✬s ❡①❡r❝✐s❡s • ✷✳✶✶✿ ❉❡r❛♥❞♦♠✐③✐♥❣ ❚❤❡♦r❡♠ ✷✳✹ • ✷✳✶✸✿ ❖♥❡ ♦❢ ❋❡✇ ❆❧✇❛②s ❲♦r❦s • ✷✳✶✺✿ ❆❧❧ ✐♥ ✸✕❛t ▼♦st ✼ ✐♥ ✶✵ • ✷✯✳✶✿ ❘❛♥❞♦♠ ❙❛♠♣❧✐♥❣ ❉♦❡s♥✬t ❲♦r❦ • ■♥ ❈❧❛ss✿ ✸✳✶✿ ❉❡❝✐s✐♦♥ ✈s✳ ❈♦♥str✉❝t✐♦♥ • ■♥ ❈❧❛ss✿ ✸✳✸✿ ❙❛t✐s✜❛❜❧❡ ✇✐t❤ ♠❛♥② ❈❧❛✉s❡s • ❆❞❞✐t✐♦♥❛❧✿ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷✳✼ P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✷ ❈♦♥s✐❞❡r s✉❜♠✐tt✐♥❣ ❡①❡r❝✐s❡ s♦❧✉t✐♦♥s ♦♥ ♣❛♣❡r✦ = ⇒ ❊①❝❡❧❧❡♥t ♣r❛❝t✐❝❡ ❢♦r ❡♥❞ ♦❢ t❡r♠ ❡①❛♠ P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✸ ✷✳✶✶✳ ❉❡r❛♥❞♦♠✐③✐♥❣ ❚❤❡♦r❡♠ ✷✳✹ ❚❤✐s ✇♦r❦s ❡①❛❝t❧② ❧✐❦❡ t❤❡ ❞❡r❛♥❞♦♠✐③❛t✐♦♥ ♦❢ ❚❤❡♦r❡♠ ✷✳✷✳ • s✇✐t❝❤✐♥❣ s✐❣♥s ✐❢ ✉♥✐t ❝❧❛✉s❡s ❛♣♣❡❛r ✐s ❛❧r❡❛❞② ❞❡t❡r♠✐♥✐st✐❝✳ • ❢♦r ❡✈❡r② ❢♦r♠✉❧❛✱ ✇❡ ❝❛♥ ❝❛❧❝✉❧❛t❡ t❤❡ ❡①♣❡❝t❡❞ s❛t✐s✜❡❞ ✇❡✐❣❤t ✐❢ ❡❛❝❤ ✈❛r✐❛❜❧❡ ✐s ♦♥❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② Φ • ❛ss✐❣♥ t❤❡ ✈❛r✐❛❜❧❡s ♦♥❡ ❜② ♦♥❡✱ ❡❛❝❤ t✐♠❡ ❛ss✐❣♥ ❛ ✈❛❧✉❡ s♦ ❛s t♦ ♠❛①✐♠✐③❡ t❤❡ ❡①♣❡❝t❛t✐♦♥ P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✹ ✷✳✶✸✳ ❖♥❡ ♦❢ ❋❡✇ ❆❧✇❛②s ❲♦r❦s ❆ ❍❛❞❛♠❛r❞✲▼❛tr✐① H ∈ {− 1 , 1 } n × n ✐s ❛ ♠❛tr✐① ✐♥ ✇❤✐❝❤ ❛❧❧ r♦✇s ❛r❡ ♣❛✐r✇✐s❡ ♦rt❤♦❣♦♥❛❧ ❛♥❞ ❛❧❧ ❝♦❧✉♠♥s ❛r❡ ♣❛✐r✇✐s❡ ♦rt❤♦❣♦♥❛❧✳ ❋♦r n ❛ ♣♦✇❡r ♦❢ t✇♦✱ t❤❡s❡ ❝❛♥ ❜❡ ❡❛s✐❧② ❝♦♥str✉❝t❡❞ ❜② st❛rt✐♥❣ ✇✐t❤ H 1 := (1) ❛♥❞ ❜② r❡❝✉rs✐✈❡❧② ❧❡tt✐♥❣ � � H n H n H 2 n := . H n − H n ◆♦t ❢♦r ❛❧❧ n ❦♥♦✇♥✱ ❡✳❣✳ n = 668 ✳ ❊✈❡r② ❍❛❞❛♠❛r❞ ♠❛tr✐① ❤❛s ♦r❞❡r ✶✱✷✱ ♦r ❛ ♠✉❧t✐♣❧❡ ♦❢ ✹✳ P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✺ ■❢ H n ∈ {− 1 , 1 } n , n ≥ 4 , ✐s ❛ ❍❛❞❛♠❛r❞ ♠❛tr✐① ✇✐t❤ t❤❡ ▲❡♠♠❛✳ ✜rst ❝♦❧✉♠♥ ❝♦♥t❛✐♥✐♥❣ ♦♥❧② ♦♥❡s✱ t❤❡♥ ❢♦r ❛♥② 1 < i, j ≤ n ✱ t❤❡r❡ ✐s ❛♥ ❡q✉❛❧ ♥✉♠❜❡r ♦❢ r♦✇s k s✉❝❤ t❤❛t ( { H n } ki , { H n } kj ) ❡q✉❛❧s (1 , 1) ✱ (1 , − 1) ✱ ( − 1 , 1) ❛♥❞ ( − 1 , − 1) ✳ Pr♦♦❢✳ ❋✐① i ❛♥❞ j ✳ ■❢ n ++ , n + − , n − + , n −− ❛r❡ t❤❡ ♥✉♠❜❡rs ♦❢ r♦✇s ✇❤❡r❡ ( { H n } ki , { H n } kj ) ❡q✉❛❧s (1 , 1) ✱ (1 , − 1) ✱ ( − 1 , 1) ❛♥❞ ( − 1 , − 1) ✱ r❡s♣❡❝t✐✈❡❧②✱ t❤❡♥ ✇❡ ❤❛✈❡ n ++ + n + − = n/ 2 ✭❜❡❝❛✉s❡ ❝♦❧✉♠♥s ✶ ❛♥❞ i ❛r❡ ♦rt❤♦❣♦♥❛❧✮✱ n ++ + n − + = n/ 2 ✭❜❡❝❛✉s❡ ❝♦❧✉♠♥s ✶ ❛♥❞ j ❛r❡ ♦rt❤♦❣♦♥❛❧✮ ❛♥❞ n ++ + n −− = n/ 2 ✭❜❡❝❛✉s❡ ❝♦❧✉♠♥s i ❛♥❞ j ❛r❡ ♦rt❤♦❣♦♥❛❧✮✳ ❋✐♥❛❧❧②✱ n ++ + n + − + n − + + n −− = n ✳ ❖♥❧② n ++ = n + − = n − + = n −− = n/ 4 s❛t✐s✜❡s ❛❧❧ t❤❡s❡ ❝♦♥str❛✐♥ts✳ ✷ P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✻ ✷✳✶✸✳ ❖♥❡ ♦❢ ❋❡✇ ❆❧✇❛②s ❲♦r❦s ✭✷✮ ❈❤♦♦s❡ t t♦ ❜❡ t❤❡ s♠❛❧❧❡st ♣♦✇❡r ♦❢ t✇♦ s✉❝❤ t❤❛t t ≥ n + 1 ✳ ◆♦t❡ t❤❛t t ❝❛♥ ❜❡ ❝❤♦s❡♥ t♦ ❜❡ ❛t ♠♦st 2 n ✳ ❈♦♥str✉❝t t❤❡ ❍❛❞❛♠❛r❞ ♠❛tr✐① H t ❛♥❞ ❝❤♦♦s❡ ♦♥❡ ♦❢ t❤❡ t r♦✇s ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✳ ■♥t❡r♣r❡t t❤❡ ✈❛❧✉❡s ❢r♦♠ t❤✐s r♦✇ ✐♥ t❤❡ ❝♦❧✉♠♥s 2 t❤r♦✉❣❤ n + 1 ❛s ❛ss✐❣♥♠❡♥ts t♦ t❤❡ n ✈❛r✐❛❜❧❡s x 1 t❤r♦✉❣❤ x n r❡s♣❡❝t✐✈❡❧②✳ ❇② t❤❡ ♣r❡✈✐♦✉s ▲❡♠♠❛✱ ❡❛❝❤ 2 ✲❝❧❛✉s❡ C ∈ F ❤❛s ❛ ♣r♦❜❛❜✐❧✐t② ♦❢ 3 4 ♦❢ ❜❡✐♥❣ s❛t✐s✜❡❞✳ ❚❤❡r❡❢♦r❡ ♦♥❡ ♦❢ t❤❡ t ❛ss✐❣♥♠❡♥ts ✭❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ r♦✇s ♦❢ H t ✮ ♠✉st ❛❧✇❛②s s❛t✐s❢② ❛t ❧❡❛st 3 4 ♠❛♥② ❝❧❛✉s❡s ♦❢ F ✳ P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✼ ✷✳✶✺✿ ❆❧❧ ✐♥ ✸✕❛t ▼♦st ✼ ✐♥ ✶✵ F := {{ a } , { b } , { c } , { d } , { a, x } , { b, x } , { c, d, x } , { c, x } , { d, x } , { a, b, x } } � �� � � �� � � �� � G 21 G 22 G 1 � �� � G 2 P❛rt✐t✐♦♥ t❤❡ ❝❧❛✉s❡s ♦❢ F ✐♥t♦ t✇♦ ❣r♦✉♣s✿ G 1 ❛♥❞ G 2 ✱ ✇❤❡r❡ G 1 ❝♦♥t❛✐♥s ❛❧❧ ❝❧❛✉s❡s ✇✐t❤ ♦♥❧② ♥❡❣❛t✐✈❡ ❧✐t❡r❛❧s ❛♥❞ G 2 ❝♦♥t❛✐♥s ❛❧❧ ❝❧❛✉s❡s ✇✐t❤ ❛t ❧❡❛st ♦♥❡ ♣♦s✐t✐✈❡ ❧✐t❡r❛❧✳ ❈❧❡❛r❧②✱ ❛♥② tr✐♣❧❡ ♦❢ ❝❧❛✉s❡s ❢r♦♠ G 1 ♦r ❢r♦♠ G 2 ❛r❡ s❛t✐s✜❛❜❧❡✳ ❆ tr✐♣❧❡ ❝♦♥t❛✐♥✐♥❣ 2 ❝❧❛✉s❡s ❢r♦♠ G 1 ❛♥❞ 1 ❢r♦♠ G 2 ❝❛♥ ❜❡ s❛t✐s✜❡❞ ❜② s❡tt✐♥❣ x �→ 0 ❛♥❞ ✉s✐♥❣ t❤❡ ❛❞❞✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡ ❢r♦♠ t❤❡ ❝❧❛✉s❡ ❢r♦♠ G 2 t♦ s❛t✐s❢② t❤❡ ❝❧❛✉s❡ ❢r♦♠ G 2 ✳ P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✽ ◆♦✇ ✇❡ ❤❛✈❡ ❛ tr✐♣❧❡ ♦❢ ❝❧❛✉s❡s ❝♦♥t❛✐♥✐♥❣ 1 ❢r♦♠ G 1 ❛♥❞ 2 ❢r♦♠ G 2 ✳ ❲❡ s♣❧✐t G 2 ✐♥t♦ t✇♦ ♣❛rts✿ G 21 ❛♥❞ G 22 ✇❤❡r❡ G 21 := {{ a } , { b } , { c } , { d }} ❛♥❞ G 22 := {{ a, x } , { b, x } , { c, x }} ✱ ❛♥❞ ❝♦♥s✐❞❡r s✉❜❝❛s❡s✿ • 2 ❢r♦♠ G 21 ✿ ❙❡t x �→ 0 ❛♥❞ s❛t✐s❢② t❤❡ t✇♦ ❝❧❛✉s❡s ❢r♦♠ G 21 ✉s✐♥❣ t❤❡ ✈❛r✐❛❜❧❡s a, b, c, d ✳ • 2 ❢r♦♠ G 22 ✿ ❙❡t x �→ 1 ❛♥❞ s❛t✐s❢② t❤❡ ❝❧❛✉s❡ ❢r♦♠ G 1 ✉s✐♥❣ ♦♥❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s a, b, c, d ✳ • 1 ❢r♦♠ ❡❛❝❤ ♦❢ G 21 ❛♥❞ G 22 ✿ ■❢ ✇❡ ♣✐❝❦❡❞ { c } ♦r { d } ❢r♦♠ G 21 t❤❡♥ ✇❡ s❡t x �→ 0 ♦t❤❡r✇✐s❡ ✇❡ s❡t x �→ 1 ✱ ❛♥❞ t❤✐s ❝❛♥ ❜❡ s❡❡♥ t♦ ❧❡❛❞ t♦ ❛ s❛t✐s❢②✐♥❣ ❛ss✐❣♥♠❡♥t ✭❱❡r✐❢②✦✮✳ P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐