Guessing more sets

Matet, Pierre

doi:10.1016/j.apal.2015.05.001

Cited by 9 publications

(7 citation statements)

References 29 publications

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“…This section is concerned with density numbers which will play an important role in the remainder of the paper. Definition Given two infinite cardinals

τ \leq σ

d (τ, σ)

denotes the least cardinality of any

X \subseteq {[σ]}^{τ}

with the property that for any

e \in {[σ]}^{τ}

, there is

x \in X

with

x \subseteq e

. Fact (Kojman , Matet ) Let

τ \leq σ

be two infinite cardinals. Then the following hold: …”

Section: Density Numbersmentioning

confidence: 99%

“…Thus, given two infinite cardinals

τ \leq σ

d (τ, σ) = σ

if and only if

cf (σ) \neq cf (τ)

and

d (τ, χ) \leq σ

for any cardinal χ with

τ \leq χ < σ

. Fact (Matet ) Let

τ \leq σ

be two infinite cardinals. Suppose that

χ^{< τ} \leq σ < χ^{τ}

for some cardinal χ.…”

Section: Density Numbersmentioning

confidence: 99%

“…In particular,

d (ω, σ) = σ^{ℵ_{0}}

for any infinite cardinal σ. Fact (Matet ) Let

τ < σ

be two infinite cardinals. Then

d (τ, σ) \leq max (d (τ, τ), cov (σ, τ^{+}, τ^{+}, cf (τ)))

. Proposition Let τ and σ be two infinite cardinals such that

τ \leq σ < τ^{+ cf (τ)}

.…”

Section: Density Numbersmentioning

confidence: 99%

“…Then

d (τ, σ) = max (d (τ, τ), σ)

. Proof By Facts and .

□

Definition Given two infinite cardinals

τ \leq σ

{MAD}_{τ, σ}

denotes the collection of all

Q \subseteq {[σ]}^{τ}

such that (i)

| a \cap b | < τ

for any two distinct members

a, b

of Q , and (ii) for any

c \in {[σ]}^{τ}

, there is

a \in Q

with

| a \cap c | = τ

. Fact (Matet ) Let

τ \leq σ

be two infinite cardinals. Then the following hold: (i)

d (τ, σ) \geq | Q |

for all

Q \in {MAD}_{τ, σ}

. (ii)If

d (τ, τ) < d (τ, σ)

, then

| Q | = d (τ, σ)

for all

Q \in {MAD}_{τ, σ}

. Proposition Let

τ \leq σ

be two infinite cardinals.…”

Section: Density Numbersmentioning

confidence: 99%

“…Then

d (τ, σ) = d (cf (τ), σ)

. Proof By Propositions and .

□

Fact (Matet ) Let σ be a singular cardinal such that

d (cf (σ), σ) = σ^{+}

. Then

pp (σ) = σ^{+}

.…”

Section: Density Numbersmentioning

confidence: 99%

See 4 more Smart Citations

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Matet

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“…This section is concerned with density numbers which will play an important role in the remainder of the paper. Definition Given two infinite cardinals

τ \leq σ

d (τ, σ)

denotes the least cardinality of any

X \subseteq {[σ]}^{τ}

with the property that for any

e \in {[σ]}^{τ}

, there is

x \in X

with

x \subseteq e

. Fact (Kojman , Matet ) Let

τ \leq σ

be two infinite cardinals. Then the following hold: …”

Section: Density Numbersmentioning

confidence: 99%

“…Thus, given two infinite cardinals

τ \leq σ

d (τ, σ) = σ

if and only if

cf (σ) \neq cf (τ)

and

d (τ, χ) \leq σ

for any cardinal χ with

τ \leq χ < σ

. Fact (Matet ) Let

τ \leq σ

be two infinite cardinals. Suppose that

χ^{< τ} \leq σ < χ^{τ}

for some cardinal χ.…”

Section: Density Numbersmentioning

confidence: 99%

“…In particular,

d (ω, σ) = σ^{ℵ_{0}}

for any infinite cardinal σ. Fact (Matet ) Let

τ < σ

be two infinite cardinals. Then

d (τ, σ) \leq max (d (τ, τ), cov (σ, τ^{+}, τ^{+}, cf (τ)))

. Proposition Let τ and σ be two infinite cardinals such that

τ \leq σ < τ^{+ cf (τ)}

.…”

Section: Density Numbersmentioning

confidence: 99%

“…Then

d (τ, σ) = max (d (τ, τ), σ)

. Proof By Facts and .

□

Definition Given two infinite cardinals

τ \leq σ

{MAD}_{τ, σ}

denotes the collection of all

Q \subseteq {[σ]}^{τ}

such that (i)

| a \cap b | < τ

for any two distinct members

a, b

of Q , and (ii) for any

c \in {[σ]}^{τ}

, there is

a \in Q

with

| a \cap c | = τ

. Fact (Matet ) Let

τ \leq σ

be two infinite cardinals. Then the following hold: (i)

d (τ, σ) \geq | Q |

for all

Q \in {MAD}_{τ, σ}

. (ii)If

d (τ, τ) < d (τ, σ)

, then

| Q | = d (τ, σ)

for all

Q \in {MAD}_{τ, σ}

. Proposition Let

τ \leq σ

be two infinite cardinals.…”

Section: Density Numbersmentioning

confidence: 99%

“…Then

d (τ, σ) = d (cf (τ), σ)

. Proof By Propositions and .

□

Fact (Matet ) Let σ be a singular cardinal such that

d (cf (σ), σ) = σ^{+}

. Then

pp (σ) = σ^{+}

.…”

Section: Density Numbersmentioning

confidence: 99%

See 3 more Smart Citations

Two‐cardinal diamond star

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2014

Mathematical Logic Qtrly

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Guessing more sets

Cited by 9 publications

References 29 publications

Two‐cardinal diamond star

Two‐cardinal diamond star

Meeting numbers and pseudopowers

Towers and clubs

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