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❉✐r❡❝t✐♦♥❛❧✐t② ❛♥❞ ♥♦♥✲❢♦r❦✐♥❣ s♣❡❝tr❛

■t❛② ❑❛♣❧❛♥

❯♥✐✈❡rs✐tät ❑♦♥st❛♥③

▲♦❣✐❝ ❈♦❧❧♦q✉✐✉♠ ✷✵✶✶ ✱ ❇❛r❝❡❧♦♥❛✱ ✶✶t❤ ✕ ✶✻t❤ ❏✉❧② ✷✵✶✶

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✶ ✴ ✷✷

❈♦♥t❡♥ts

■♥tr♦❞✉❝t✐♦♥ t♦ ◆■P ❛♥❞ ❉✐r❡❝t✐♦♥❛❧✐t② ❚r✐❝❤♦t♦♠② t❤❡♦r❡♠✳ ❉❡✜♥❛❜✐❧✐t②✳ ▼✉❧t✐♣❧❡ ♦r❞❡r ♣r♦♣❡rt② ❛♥❞ ❛ q✉❡st✐♦♥ ♦❢ ❉❡❧♦♥✳ ◆♦♥✲s♣❧✐♥t❡r✐♥❣✳ ❖♣❡♥ q✉❡st✐♦♥s✳ ■♥tr♦❞✉❝t✐♦♥ t♦ ◆♦♥✲❢♦r❦✐♥❣ s♣❡❝tr❛✳ ❙♦♠❡ t❤❡♦r❡♠s✳ ❙♦♠❡ ❡①❛♠♣❧❡s✳

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✷ ✴ ✷✷

◆■P ❛♥❞ ❉✐r❡❝t✐♦♥❛❧✐t②

❚❤✐s ♣❛rt ✐s ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❙❛❤❛r♦♥ ❙❤❡❧❛❤✳

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✸ ✴ ✷✷

◆■P ❛♥❞ ❉✐r❡❝t✐♦♥❛❧✐t②

❇② P♦✐③❛t ❛♥❞ ❙❤❡❧❛❤✱ ❛ t❤❡♦r② ❚ ✐s ❞❡♣❡♥❞❡♥t ✐✛ ❣✐✈❡♥ ❛ t②♣❡ ♣ ∈ ❙ (▼) t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♠♣❧❡t❡ ❣❧♦❜❛❧ t②♣❡s q ∈ ❙ (C) t❤❛t ❡①t❡♥❞ ✐t ❛♥❞ ❛r❡ ✜♥✐t❡❧② s❛t✐s✜❛❜❧❡ ✐♥ ▼ ✐s ❛t ♠♦st ✷|▼| ✭✇❤✐❧❡ t❤❡ ♠❛①✐♠❛❧ ♥✉♠❜❡r ✐s ✷✷|▼|✮✳

❉❡✜♥✐t✐♦♥

❋♦r ❛ t②♣❡ ♣ (¯ ①) ∈ ❙ (❆)✱ ❧❡t ✉❢ (♣) = {q ∈ ❙ (C) |q ✐s ❢✳s✳ ♦♥ ❆ ❛♥❞ q ⊇ ♣} . ❋♦r ❛ ♣❛rt✐❛❧ t②♣❡ ♣ (¯ ①) ♦✈❡r ❛ s❡t ❆✱ ❛♥❞ ❛ s❡t ♦❢ ❢♦r♠✉❧❛s ∆✱ ✉❢∆ (♣) = {q (¯ ①) ∈ ❙∆ (C) |q ∪ ♣ ✐s ❢✳s✳ ♦♥ ❆} .

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✹ ✴ ✷✷

❉✐r❡❝t✐♦♥❛❧✐t②

❉❡✜♥✐t✐♦♥

▲❡t ❚ ❜❡ ❛♥② t❤❡♦r② t❤❡♥✿

✶ ❚ ✐s s❛✐❞ t♦ ❜❡ ♦❢ ❜♦✉♥❞❡❞ ❞✐r❡❝t✐♦♥❛❧✐t② ✭♦r ❥✉st✱ ❚ ✐s ❜♦✉♥❞❡❞✮ ✐❢ ❢♦r

❛❧❧ ▼ | = ❚✱ ♣ ∈ ❙ (▼)✱ |✉❢ (♣)| ≤ ✷|❚| ✳

✷ ❚ ✐s s❛✐❞ t♦ ❜❡ ♦❢ ♠❡❞✐✉♠ ❞✐r❡❝t✐♦♥❛❧✐t② ✭♦r ❥✉st✱ ❚ ✐s ♠❡❞✐✉♠✮ ✐❢ ❢♦r

❛❧❧ ▼ | = ❚✱ ♣ ∈ ❙ (▼)✱ |✉❢ (♣)| ≤ |▼||❚| ❛♥❞ ❚ ✐s ♥♦t ❜♦✉♥❞❡❞✳

✸ ❚ ✐s s❛✐❞ t♦ ❜❡ ♦❢ ❧❛r❣❡ ❞✐r❡❝t✐♦♥❛❧✐t② ✭♦r ❥✉st✱ ❚ ✐s ❧❛r❣❡✮ ✐❢ ❚ ✐s ♥♦t

❜♦✉♥❞❡❞ ♥♦r ♠❡❞✐✉♠✳

❋❛❝t

■❢ ❚ ❤❛s t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ♣r♦♣❡rt②✱ t❤❡♥ ✐t ✐s ❧❛r❣❡✳

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✺ ✴ ✷✷

❉✐r❡❝t✐♦♥❛❧✐t②

❉❡✜♥✐t✐♦♥

❋♦r ❛ ❝❛r❞✐♥❛❧ λ ❧❡t ❉❡❞∗ (λ) ❜❡ t❤❡ ✜rst ❝❛r❞✐♥❛❧ µ s✉❝❤ t❤❛t t❤❡r❡ ✐s ♥♦ s❡t t❤❡♦r❡t✐❝ tr❡❡ ✭✐✳❡✳ ❛ ♣❛rt✐❛❧ ♦r❞❡r s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ①✱ t❤❡ s❡t ♦❢ ❡❧❡♠❡♥ts s♠❛❧❧❡r t❤❛♥ ① ✐s ✇❡❧❧✲♦r❞❡r❡❞✮ ✇✐t❤ ≤ λ ♠❛♥② ♥♦❞❡s ❛♥❞ ≥ µ ❜r❛♥❝❤❡s ♦❢ t❤❡ s❛♠❡ ❤❡✐❣❤t✳ ▲✐❦❡✇✐s❡✱ ❧❡t ❉❡❞ (λ) ❜❡ t❤❡ ✜rst ❝❛r❞✐♥❛❧ µ s✉❝❤ t❤❛t t❤❡r❡ ✐s ♥♦ ❧✐♥❡❛r❧② ♦r❞❡r❡❞ s❡t ♦❢ ❝❛r❞✐♥❛❧✐t② µ ✇✐t❤ ❛ ❞❡♥s❡ s✉❜s❡t ♦❢ ❝❛r❞✐♥❛❧✐t② λ✳ ▲❡t ❞❡❞ (λ) ❜❡ t❤❡ ♣r❡❞❡❝❡ss♦r ♦❢ ❉❡❞ (λ) ✐❢ ✐t ❡①✐sts ♦r ❉❡❞ (λ) ✐❢ ♥♦t✱ ❛♥❞ ❧✐❦❡✇✐s❡✱ ❧❡t ❞❡❞∗ (λ) ❜❡ t❤❡ ♣r❡❞❡❝❡ss♦r ♦❢ ❉❡❞∗ (λ) ✐❢ ✐t ❡①✐sts ❛♥❞ ❉❡❞∗ (λ) ✐❢ ♥♦t✳

❋❛❝t

λ+ < ❉❡❞∗ (λ) ≤ ❉❡❞ (λ) ≤

• ✷λ+ ❛♥❞ λ < ❞❡❞∗ (λ) ≤ ❞❡❞ (λ) ≤ ✷λ✳

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✻ ✴ ✷✷

❉✐r❡❝t✐♦♥❛❧✐t②

❚❤❡♦r❡♠

✭❚ ❞❡♣❡♥❞❡♥t✮

✶ ❚ ✐s ❜♦✉♥❞❡❞ ✐✛ ❢♦r ❛❧❧ ✜♥✐t❡ ∆✱ ▼ |

= ❚ ❛♥❞ ♣ ∈ ❙ (▼)✱ ✉❢∆ (♣) ✐s ✜♥✐t❡✳

✷ ❚ ✐s ♠❡❞✐✉♠ ✐✛ ❢♦r ❡✈❡r② λ ≥ |❚|✱

λ = s✉♣ {|✉❢∆ (♣)| |♣ ∈ ❙ (▼) , ∆ ✜♥✐t❡, |▼| = λ} .

✸ ❚ ✐s ❧❛r❣❡ ✐✛ ❢♦r ❡✈❡r② λ ≥ |❚|✱

❞❡❞∗ (λ) ≤ s✉♣ {|✉❢∆ (♣)| |♣ ∈ ❙ (▼) , ∆ ✜♥✐t❡, |▼| = λ} ≤ ❞❡❞ (λ) .

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✼ ✴ ✷✷

❉✐r❡❝t✐♦♥❛❧✐t②

❉❡✜♥✐t✐♦♥

▲❡t ▼♣ ❜❡ ▼ ❡♥r✐❝❤❡❞ ✇✐t❤ ❡①t❡r♥❛❧❧② ❞❡✜♥❛❜❧❡ s❡ts ❞❡✜♥❡❞ ♦✈❡r ❛ r❡❛❧✐③❛t✐♦♥ ¯ ❛ | = ♣✳

❈❧❛✐♠

✶ ❙✉♣♣♦s❡ ❚ ✐s ♥♦t ❧❛r❣❡✱ ♣ ∈ ❙ (▼)✱ q ∈ ✉❢ (♣)✱ ❛♥❞

¯ ❛ = ❛✵, ❛✶, . . . | = q(ω) ↾ ▼✳ ❚❤❡♥✱ t♣ (¯ ❛/▼) ✐s ❞❡✜♥❛❜❧❡ ✇✐t❤ ♣❛r❛♠❡t❡rs ✐♥ ▼♣✳

✷ ■❢ ❚ ✐s ❜♦✉♥❞❡❞ t❤❡♥ t♣ (¯

❛/▼) ✐s ❞❡✜♥❛❜❧❡ ♦✈❡r ❛❝❧❡q (∅) ✐♥ ▼♣✳

✸ ✭■❢ ❚ ✐s st❛❜❧❡ t❤❡♥ t♣ (¯

❛/▼) ✐s ❞❡✜♥❛❜❧❡ ♦✈❡r ❞❝❧❡q (∅) ✐♥ ▼♣✳✮

◗✉❡st✐♦♥

■s t❤❡ ❞❡✜♥✐t✐♦♥ ❛❜♦✈❡ q✉❛♥t✐✜❡r ❢r❡❡❄

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✽ ✴ ✷✷

▼✉❧t✐♣❧❡ ♦r❞❡r ♣r♦♣❡rt②

❉❡✜♥✐t✐♦♥

❆ t❤❡♦r② ❚ ✐s s❛✐❞ t♦ ❤❛✈❡ t❤❡ ♠✉❧t✐♣❧❡ ♦r❞❡r ♣r♦♣❡rt② ✐❢ t❤❡r❡ ❛r❡ ❢♦r♠✉❧❛s ϕ♥ (①, ②♥) ❢♦r ♥ < ω s✉❝❤ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ♦❢ ❢♦r♠✉❧❛s ✐s ❝♦♥s✐st❡♥t ✇✐t❤ ❚✿ Γ =

• ϕ♥
• ①η, ②♥,❦

η(❦)<♥ |η ∈ ωω

• .

❈❧❛✐♠

■❢ ❚ ✐s str♦♥❣❧② ❞❡♣❡♥❞❡♥t✱ ❢♦r ❡①❛♠♣❧❡✱ ✐❢ ❚ = ❘❈❋✱ ✐t ❞♦❡s ♥♦t ❤❛✈❡ t❤❡ ♠✉❧t✐♣❧❡ ♦r❞❡r ♣r♦♣❡rt②✳

❋❛❝t

❬❑❡✐s❧❡r❪ ■❢ ❚ ✐s ❝♦✉♥t❛❜❧❡ ❛♥❞ ❤❛s t❤❡ ♠✉❧t✐♣❧❡ ♦r❞❡r ♣r♦♣❡rt②✱ t❤❡♥ ❢♦r ❡✈❡r② κ✱ s✉♣ {❙ (▼) |▼ | = ❚, |▼| = κ} ≥ (❞❡❞ (κ))ω✳ ■❢ ❚ ❞♦❡s ♥♦t ❤❛✈❡ t❤❡ ♠✉❧t✐♣❧❡ ♦r❞❡r ♣r♦♣❡rt②✱ t❤❡♥ s✉♣ {❙ (▼) |▼ | = ❚, |▼| = κ} ≤ ❞❡❞ (κ) ❢♦r ❡✈❡r② κ✳

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✾ ✴ ✷✷

▼✉❧t✐♣❧❡ ♦r❞❡r ♣r♦♣❡rt②

◗✉❡st✐♦♥

❬❉❡❧♦♥❪ ▼❛✐s ✐❧ ❧❛✐ss❡ ♦✉✈❡rt❡ ❧❛ ♣♦ss✐❜✐❧✐té q✉❡ ❧❛ ❜♦r♥❡ ❞✉ ♥♦♠❜r❡ ❞❡ ❝♦❤ér✐t✐❡rs s♦✐t ❞❡❞ (|▼|) ❞❛♥s ❧❡ ❝❛s ❞❡ ❧❛ ♣r♦♣r✐été ❞❡ ❧✬♦r❞r❡ ❡t (❞❡❞ (|▼|))(ω) ❞❛♥s ❧❡ ❝❛s ❞❡ ❧✬♦r❞r❡ ♠✉❧t✐♣❧❡✳ ❚r❛♥s❧❛t✐♦♥✿ ■s t❤❡r❡ ❛ ❝♦✉♥t❛❜❧❡ t❤❡♦r② ✇✐t❤♦✉t t❤❡ ♠✉❧t✐♣❧❡ ♦r❞❡r ♣r♦♣❡rt② s✉❝❤ t❤❛t ❢♦r ❡✈❡r② κ ≥ ℵ✵✱ s✉♣ {|✉❢ (♣)| |♣ ∈ ❙ (▼) , |▼| = κ} = (❞❡❞ (κ))ω✳

❚❤❡♦r❡♠

❘❈❋ ❤❛s ❧❛r❣❡ ❞✐r❡❝t✐♦♥❛❧✐t②✳ ▼♦r❡♦✈❡r✱ ❘❈❋ ❞♦❡s ♥♦t ❤❛✈❡ t❤❡ ♠✉❧t✐♣❧❡ ♦r❞❡r ♣r♦♣❡rt② ❜✉t ❢♦r ❡✈❡r② λ ≥ ℵ✵✱ s✉♣ {|✉❢ (♣)| |♣ ∈ ❙ (▼) , |▼| = λ} ≥ (❞❡❞ (λ))ω .

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✶✵ ✴ ✷✷

❙♣❧✐♥t❡r✐♥❣

❉❡✜♥✐t✐♦♥

▲❡t ♣ ∈ ❙ (▼)✳ ❙❛② t❤❛t ♣ s♣❧✐ts ♦✈❡r ❛ s❡t ❆ ✐❢ t❤❡r❡ ✐s ❛ ❢♦r♠✉❧❛ ϕ (①, ②) ❛♥❞ ❛, ❜ ∈ ▼ s✉❝❤ t❤❛t ❛ ≡❆ ❜ ❛♥❞ ϕ (①, ❛) ∧ ¬ϕ (①, ❜) ∈ ♣✳

❉❡✜♥✐t✐♦♥

❬●r♦ss❜❡r❣✱ ❱✐❧❧❛✈❡❝❡s✱ ❱❛♥❉✐❡r❡♥❪ ▲❡t ♣ ∈ ❙ (C)✳ ❙❛② t❤❛t ♣ s♣❧✐♥t❡rs ♦✈❡r ▼ ✐❢ t❤❡r❡ ✐s s♦♠❡ σ ∈ ❆✉t (C) s✉❝❤ t❤❛t

✶ σ (♣) = ♣✳ ✷ σ (♣|▼) = ♣|▼✳ ✸ σ (▼) = ▼ s❡t✇✐s❡✳

❋❛❝t

■❢ ❚ ✐s st❛❜❧❡✱ t❤❡♥ s♣❧✐♥t❡r✐♥❣ ✐s t❤❡ s❛♠❡ ❛s s♣❧✐tt✐♥❣✳

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✶✶ ✴ ✷✷

❙♣❧✐♥t❡r✐♥❣

❚❤❡♦r❡♠

❙✉♣♣♦s❡ ❚ ❤❛s ♠❡❞✐✉♠ ♦r ❧❛r❣❡ ❞✐r❡❝t✐♦♥❛❧✐t② ✭t❤✐s ✐♥❝❧✉❞❡s t❤❡ ♦♣t✐♦♥ t❤❛t ❚ ❤❛s ■P✮✳ ❚❤❡♥ s♣❧✐tt✐♥❣ = s♣❧✐♥t❡r✐♥❣✳

❚❤❡♦r❡♠

❚❤❡r❡ ✐s ❛ t❤❡♦r② ✇✐t❤ ❜♦✉♥❞❡❞ ❞✐r❡❝t✐♦♥❛❧✐t② ✐♥ ✇❤✐❝❤ s♣❧✐tt✐♥❣ = s♣❧✐♥t❡r✐♥❣✳

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✶✷ ✴ ✷✷

❖♣❡♥ ◗✉❡st✐♦♥s

▲❡♠♠❛

✭❚ ❞❡♣❡♥❞❡♥t✮ ❙✉♣♣♦s❡ ∆ ✐s ❛ ✜♥✐t❡ s❡t ♦❢ ❢♦r♠✉❧❛s✱ ¯ ① ❛ ✜♥✐t❡ t✉♣❧❡ ♦❢ ✈❛r✐❛❜❧❡s✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ♥ < ω ❛♥❞ ✜♥✐t❡ s❡t ♦❢ ❢♦r♠✉❧❛s ∆✵ s✉❝❤ t❤❛t ❢♦r ❡✈❡r② s❡t ❆✱ ✐❢ q✶, q✷ ∈ ❙¯

① (C) ❛r❡ ❢✳s✳ ✐♥ ❆ ❛♥❞

• q(♥)

✶

↾ ∆✵

• |❆ =
• q(♥)

✷

↾ ∆✵

• |❆ t❤❡♥ q✶ ↾ ∆ = q✷ ↾ ∆✳

◗✉❡st✐♦♥

■s t❤✐s ❧❡♠♠❛ tr✉❡ ❢♦r ✐♥✈❛r✐❛♥t t②♣❡s❄ ✭❛♥❞ ♥♦t ❥✉st ✜♥✐t❡❧② s❛t✐s✜❛❜❧❡✮✳

❚❤❡♦r❡♠

■❢ ❚ ✐s ❛♥ ♦✲♠✐♥✐♠❛❧ ❡①♣❛♥s✐♦♥ ♦❢ t❤❡ r❡❛❧s✱ t❤❡♥ ❚ ❤❛s ❧❛r❣❡ ❞✐r❡❝t✐♦♥❛❧✐t②✳

◗✉❡st✐♦♥

■s ❛♥ ❡①♣❛♥s✐♦♥ ♦❢ ❛ ❧❛r❣❡ t❤❡♦r② st✐❧❧ ❧❛r❣❡❄

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✶✸ ✴ ✷✷

◆♦♥✲❢♦r❦✐♥❣ ❙♣❡❝tr❛

❚❤✐s ♣❛rt ✐s ❥♦✐♥t ✇♦r❦ ✭✐♥ ♣r♦❣r❡ss✮ ✇✐t❤ ❆rt❡♠ ❈❤❡r♥✐❦♦✈ ❛♥❞ ❙❛❤❛r♦♥ ❙❤❡❧❛❤✳

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✶✹ ✴ ✷✷

◆♦♥✲❢♦r❦✐♥❣ ❙♣❡❝tr❛

▲❡t ✉s r❡❝❛❧❧ t❤❡ ❜❛s✐❝ ❞❡✜♥✐t✐♦♥s✿

❉❡✜♥✐t✐♦♥

✭❞✐✈✐❞✐♥❣✮ ▲❡t ❆ ❜❡ ❜❡ ❛ s❡t✱ ❛♥❞ ❛ ❛ t✉♣❧❡✳ ❲❡ s❛② t❤❛t t❤❡ ❢♦r♠✉❧❛ ϕ (①, ❛) ❞✐✈✐❞❡s ♦✈❡r ❆ ✐✛ t❤❡r❡ ✐s ❛ ♥✉♠❜❡r ❦ < ω ❛♥❞ t✉♣❧❡s {❛✐ |✐ < ω} s✉❝❤ t❤❛t

✶ t♣ (❛✐/❆) = t♣ (❛/❆)✳ ✷ ❚❤❡ s❡t {ϕ (①, ❛✐) |✐ < ω} ✐s ❦✲✐♥❝♦♥s✐st❡♥t ✭✐✳❡✳ ❡✈❡r② s✉❜s❡t ♦❢ s✐③❡ ❦

✐s ♥♦t ❝♦♥s✐st❡♥t✮✳

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✶✺ ✴ ✷✷

◆♦♥✲❢♦r❦✐♥❣ ❙♣❡❝tr❛

❉❡✜♥✐t✐♦♥

✭❢♦r❦✐♥❣✮ ▲❡t ❆ ❜❡ ❜❡ ❛ s❡t✱ ❛♥❞ ❛ ❛ t✉♣❧❡✳

✶ ❙❛② t❤❛t t❤❡ ❢♦r♠✉❧❛ ϕ (①, ❛) ❢♦r❦s ♦✈❡r ❆ ✐❢ t❤❡r❡ ❛r❡ ❢♦r♠✉❧❛s

ψ✐ (①, ❛✐) ❢♦r ✐ < ♥ s✉❝❤ t❤❛t ϕ (①, ❛) ⊢

✐<♥ ψ✐ (①, ❛✐) ❛♥❞ ψ✐ (①, ❛✐)

❞✐✈✐❞❡s ♦✈❡r ❆ ❢♦r ❡✈❡r② ✐ < ♥✳

✷ ❙❛② t❤❛t ❛ t②♣❡ ♣ ❢♦r❦s ♦✈❡r ❆ ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡ ❝♦♥❥✉♥❝t✐♦♥ ♦❢

❢♦r♠✉❧❛s ❢r♦♠ ♣ ✇❤✐❝❤ ❢♦r❦s ♦✈❡r ❆✳

✸ ❚❤❡ ♥♦t❛t✐♦♥ ❛ |

⌣

❢ ❆ ❜ ♠❡❛♥s✿ t♣ (❛/❆❜) ❞♦❡s ♥♦t ❢♦r❦ ♦✈❡r ❆✳

❉❡✜♥✐t✐♦♥

❋♦r ❛ ❝♦✉♥t❛❜❧❡ t❤❡♦r② ❚ ❛♥❞ κ ≤ λ✱ ❧❡t ❢ ♥

❚ (κ, λ) ❜❡

s✉♣

• ❙♥❢

♥ (◆, ▼) |▼ ≺ ◆ |

= ❚, |▼| ≤ κ, |◆| ≤ λ

• ✱ ✇❤❡r❡

❙♥❢

♥ (❆, ❇) = {♣ ∈ ❙♥ (❆) |♣ ❞♦❡s ♥♦t ❢♦r❦ ♦✈❡r ❇ }✳ ❲❡ ❝❛❧❧ t❤✐s ❢✉♥❝t✐♦♥

t❤❡ ♥♦♥✲❢♦r❦✐♥❣ s♣❡❝tr✉♠ ♦❢ ❚✳

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✶✻ ✴ ✷✷

▼♦t✐✈❛t✐♥❣ ◗✉❡st✐♦♥

❋♦r ❛ ❝♦✉♥t❛❜❧❡ ❝♦♠♣❧❡t❡ t❤❡♦r② ❚✱ ❧❡t ❣❚ (κ) = s✉♣ {❙ (▼) ||▼| = κ} ✇❤❡r❡ ❙ (▼) ✐s t❤❡ s❡t ♦❢ t②♣❡s ✐♥ ✶ ✈❛r✐❛❜❧❡ ♦✈❡r ▼✳ ❚❤❡♥ ❜② ✇♦r❦ ♦❢ ❙❤❡❧❛❤ ❛♥❞ ❑❡✐s❧❡r✱

❋❛❝t

❚❤❡r❡ ❛r❡ ♦♥❧② s✐① ♣♦ss✐❜✐❧✐t✐❡s ❢♦r ❣❚✿ κ✱ κ + ✷ω✱ κω✱ ❞❡❞ κ✱(❞❡❞ κ)ω✱ ✷κ✳ ◆♦t❡ t❤❛t ❣❚ (κ) = ❢❚ (κ, κ)✳

◗✉❡st✐♦♥

❲❤❛t ❛r❡ t❤❡ ♣♦ss✐❜✐❧✐t✐❡s ❢♦r ❢❚❄ ❛r❡ t❤❡r❡ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② ♣♦ss✐❜✐❧✐t✐❡s❄

◗✉❡st✐♦♥

■s t❤❡r❡ ❛♥② ❞✐✛❡r❡♥❝❡ ✐♥ ❝♦✉♥t✐♥❣ ✶✲t②♣❡s ❛♥❞ ♥✲t②♣❡s❄

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✶✼ ✴ ✷✷

❙♦♠❡ ♣❛rt✐❛❧ r❡s✉❧ts

▲❡♠♠❛

❙♦♠❡ s✐♠♣❧❡ ✐♥❡q✉❛❧✐t✐❡s✿

✶ κ ≤ ❢❚ (κ, λ) ≤ ✷λ✳ ✷ ✐❢ ❢❚ (κ, λ) ≥ µ ❛♥❞ λ ≥ κ′ ≥ κ ⇒ ❢❚ (κ′, λ) ≥ µ✳ ✸ ❢ ♥

❚ (κ, λ) ≤ ❢ ♥+✶ ❚

(κ, λ)✳

❋❛❝t

❋♦r st❛❜❧❡ ❚✱ ✇❡ ❤❛✈❡ ❢❚ (κ, λ) ≤ κℵ✵✳ ■♥ ❢❛❝t ✇❡ ❦♥♦✇ ❡①❛❝t❧② ✇❤❛t ❛r❡ t❤❡ ♣♦ss✐❜✐❧✐t✐❡s ❢♦r ❢❚✱ ♥❛♠❡❧②✿ κ✱ κ + ✷ℵ✵ ❛♥❞ κℵ✵✳

❋❛❝t

■❢ ❚ ✐s ◆■P ❛♥❞ ♥♦t st❛❜❧❡✱ t❤❡♥ ❞❡❞ (κ) ≤ ❢❚ (κ, λ) ≤ ❞❡❞ (κ)ω✳

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✶✽ ✴ ✷✷

❇♦✉♥❞❡❞ ❛♥❞ ✉♥❜♦✉♥❞❡❞

❉❡✜♥✐t✐♦♥

❲❡ s❛② ❚ ❤❛s ❜♦✉♥❞❡❞ ♥♦♥✲❢♦r❦✐♥❣ s♣❡❝tr✉♠ ✐❢ t❤❡r❡ ✐s ❛ ❢✉♥❝t✐♦♥ ❣ s✉❝❤ t❤❛t ❢❚ (κ, λ) ≤ ❣ (κ)✳

❋❛❝t

❚ ✐s ❜♦✉♥❞❡❞ ✐✛ ❚ ✐s ❜♦✉♥❞❡❞ ❜② ✷✷κ✱ ✐✳❡✳ ❢❚ (κ, λ) > ✷✷κ ❢♦r s♦♠❡ κ ≤ λ ✐✛ ❢❚ (κ, λ) ≥ λ ❢♦r ❛❧❧ κ ≤ λ✳

◗✉❡st✐♦♥

❬❆❞❧❡r❪ ❙✉♣♣♦s❡ ❚ ✐s ❜♦✉♥❞❡❞✱ ❞♦❡s ✐t ❢♦❧❧♦✇ t❤❛t ✐t ✐s ◆■P❄

❈❧❛✐♠

■❢ ❚ ✐s ❜♦✉♥❞❡❞✱ ❛♥❞ ❤❛s ■P t❤❡♥ ❢❚ (κ, λ) = ✷♠✐♥{✷κ,λ}✳

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✶✾ ✴ ✷✷

■♥s✐❞❡ ◆❚P✷

❉❡✜♥✐t✐♦♥

❲❡ s❛② ❚ ❤❛s ■P❢ ✐❢ t❤❡r❡ ✐s s♦♠❡ ❢♦r♠✉❧❛ ϕ (①, ②) ❛♥❞ s♦♠❡ ❝♦✉♥t❛❜❧❡ s❡t ❆✱ ❆ ⊆ ▼ | = ❚ ❛♥❞ ❛♥ ❆ ✐♥❞✐s❝❡r♥✐❜❧❡ s❡q✉❡♥❝❡ ❛✐ |✐ < ω ⊆ ▼ s✉❝❤ t❤❛t

• ϕ (①, ❛✐)✐ ❡✈❡♥

❞♦❡s ♥♦t ❢♦r❦ ♦✈❡r ❆✳

❚❤❡♦r❡♠

▲❡t ❚ ❜❡ ◆❚P✷✳ ❚❋❆❊

✶ ❚ ❤❛s ■P✳ ✷ ❚ ❤❛s ■P❢ ✳ ✸ ❢❚ (κ, λ) = ✷λ ❢♦r ❡✈❡r② κ ≤ λ✳ ✹ ❢❚ (κ, λ) > ❞❡❞ (κ)ω ❢♦r s♦♠❡ κ ≤ λ✳

❈♦r♦❧❧❛r②

❚ ✐s ◆■P ✐✛ ❢❚ (κ, λ) ≤ ❞❡❞ (κ)ω✳

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✷✵ ✴ ✷✷

❙♦♠❡ ❡①❛♠♣❧❡s

❚❤❡♦r❡♠

✶ ❚❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ t❤❡♦r② ❚ s✉❝❤ t❤❛t ❢ ♥

❚ (κ, λ) = ✷♠✐♥{λ,✷κ}✳ ■♥

❢❛❝t✱ ❛ t②♣❡ ❞♦❡s ♥♦t ❢♦r❦ ♦✈❡r ❆ ✐✛ ✐t ✐s ❢✳s✳ ✐♥ ❆✳

✷ ❚❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ t❤❡♦r② ❚ s✉❝❤ t❤❛t ❢ ♥

❚ (κ, λ) = λ ❢♦r ❛❧❧

✷✷κ ≤ λ✳

✸ ❚❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ t❤❡♦r② ❚ s✉❝❤ t❤❛t ❢ ✶

❚ (κ, λ) ≤ ✷♠✐♥{λ,✷κ} ❛♥❞

❢ ✷

❚ (κ, λ) = ✷λ✳

✹ ❚❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ t❤❡♦r② ❚ s✉❝❤ t❤❛t ❢ ♥

❚ (κ, λ) = λℵ✵✳

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✷✶ ✴ ✷✷

❚❤❡ ❊♥❞

❚❤❛♥❦ ❨♦✉✦

■t❛② ❑❛♣❧❛♥ ✭❯♥✐✲❑♦♥st❛♥③ ✮ ❉✐r❡❝t✐♦♥❛❧✐t②✱ ♥❢ s♣❡❝tr❛ ❏✉❧②✱ ✷✵✶✶ ✷✷ ✴ ✷✷