Cardinal characteristics associated with familiesof functions and permutations
Nattapon Sonpanow
RIMS Set Theory Workshop

November 20, 2020
10:30 – 11:00 (GMT+9)

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Cardinals associated with functions and permutations

Notations:B

A is the set of all functions from B to A.

Sym(A) is the set of all permutations on A.
[A]ω = {X ⊆ A : |X| = ℵ0 }.

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Cardinals associated with functions and permutations

Continuum Hypothesis (CH)
There is no cardinal κ such that
ℵ0 < κ < c.
In other words, ℵ1 = c.
It is well-known that CH is relatively independent from ZFC.
Some concepts in infinitary combinatorics lead to cardinal characteristics
which lie inclusively between ℵ1 and c. These cardinals are mostly definedon families of infinite sets of natural numbers. We study cardinal
characteristics associated with families of functions and permutations on
the set of natural numbers.

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Cardinals associated with functions and permutations

Continuum Hypothesis (CH)
There is no cardinal κ such thatℵ0 < κ < c.
In other words, ℵ1 = c.
It is well-known that CH is relatively independent from ZFC.
Some concepts in infinitary combinatorics lead to cardinal characteristics
which lie inclusively between ℵ1 and c. These cardinals are mostly definedon families of infinite sets of natural numbers. We study cardinal
characteristics associated with families of functions and permutations on
the set of natural numbers.

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Cardinals associated with functions and permutations

cianon(M)rdbscov(M)

p
ℵ1

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Cardinals associated with functions and permutations

The Cardinals cov(M) and non(M)Let M be the set of meagre subsets of R.
{}∪cov(M) = min |A| : A ⊆ M and
A=R ,{}
non(M) = min |A| : A ⊆ R and A ∈/M .

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Cardinals associated with functions and permutations

The Cardinal a

Two sets X, Y ∈ [ω]ω are almost disjoint if X ∩ Y is finite.
An infinite family A ⊆ [ω]ω is an almost disjoint (a.d.) family if its
members are pairwise almost disjoint. Such a family A is a maximal
almost disjoint (m.a.d.) family if it is maximal with respect to theinclusion.
a = min{|A| : A is a m.a.d. family}.

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Cardinals associated with functions and permutations

The Cardinals ae and ap

In [12], Zhang extends the concept of almost disjoint family to families offunctions and permutations on ω.
Two functions f, g ∈ ω ω are almost disjoint if f ∩ g is finite.
ae = min{|A| : A ⊆ ω ω is a m.a.d. family of functions},
ap = min{|A| : A ⊆ Sym(ω) is a m.a.d. family of permutations}.

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Cardinals associated with functions and permutations

Known Results

Theorem 1 (Zhang, 1999)
Each of the following statements is relatively consistent with ZFC:a < ae ,a < ap .
Theorem 2 (Brendle, Spinas, Zhang, 2000)
non(M) ≤ ae ,non(M) ≤ ap .

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Cardinals associated with functions and permutations

cae

apianon(M)rdbscov(M)

pℵ1

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Cardinals associated with functions and permutations

The Cardinals d and b
For any two functions f, g ∈ ω ω, we say that g dominates f iff(n) ≤ g(n) for all but finitely many n ∈ ω.
A family D ⊆ ω ω is a dominating family if each function in ω ω isdominated by some member of D.
d = min{|D| : D ⊆ ω ω is a dominating family}.
A family B ⊆ ω ω is an unbounded family if there is no function in ω ωwhich dominates every member of B.
b = min{|B| : B ⊆ ω ω is an unbounded family}.

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Cardinals associated with functions and permutations

New Cardinals dp and bp

dp = min{|D| : D ⊆ Sym(ω) is a dominating family},bp = min{|B| : B ⊆ Sym(ω) is an unbounded family}.

Theorem 3
dp = c,bp = 2.

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Cardinals associated with functions and permutations

The Cardinals s and r

For any two sets A, B ∈ [ω]ω , we say that A splits B if
B ∩ A and B \ A are infinite.
A family S ⊆ [ω]ω is a splitting family if each member of [ω]ω is split bysome member of S.
s = min{|S| : S ⊆ [ω]ω is a splitting family}.
A family R ⊆ [ω]ω is a reaping family if there is no set in [ω]ω which splits
every member of R.
r = min{|R| : R ⊆ [ω]ω is a reaping family}.

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Cardinals associated with functions and permutations

New Cardinals sf , rf , sp , and rp

sf = min{|S| : S ⊆ ω ω is a splitting family},rf = min{|R| : R ⊆ ω ω is a reaping family},sp = min{|S| : S ⊆ Sym(ω) is a splitting family},rp = min{|R| : R ⊆ Sym(ω) is a reaping family}.
Fact. s ≤ non(M) and cov(M) ≤ r.
Theorem 4 (S.)
sf = non(M) = sp ,rf = cov(M) ≤ rp .

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Cardinals associated with functions and permutations

sf = non(M) and rf = cov(M)Proof. Use the fact thatcov(M) = min{|C| : C ⊆ ω ω ∧ ¬∃f ∈ ω ω ∀g ∈ C [f ∩ g is infinite]}, andnon(M) = min{|C| : C ⊆ ω ω ∧ ∀f ∈ ω ω ∃g ∈ C [f ∩ g is infinite]}.
cov(M) ≤ rp
Proof. Use the fact that cov(M) = mctbl and consider the poset Fn1-1 (ω, ω).
non(M) = sp
Proof. Consider Sym(ω) as a Polish space.
Try to connect a set of functions and a set of permutations.

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Cardinals associated with functions and permutations

The Cardinal i

An infinite family I ⊆ [ω]ω is an independent family if, for any two finite∩∪∩
disjoint sets A, B ⊆ I, A \ B is infinite (here ∅ = ω).
Such a family I is a maximal independent family if it is maximal withrespect to the inclusion.
i = min{|I| : I ⊆ [ω]ω is a maximal independent family}.

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Cardinals associated with functions and permutations

New Cardinals if and ip

if = min{|I| : I ⊆ ω ω is a maximal independent family},
ip = min{|I| : I ⊆ Sym(ω) is a maximal independent family}.
Theorem 5 (S. and Vejjajiva, 2020)
cov(M) ≤ d ≤ if ≤ i,cov(M) ≤ ip ≤ i.
Note that, for ae and ap ,ZFC ̸⊢ ae ≤ a,
ZFC ̸⊢ ap ≤ a.

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Cardinals associated with functions and permutations

cov(M) ≤ if , ip
Proof. Use the fact that cov(M) = mctbl and consider the posets Fn(ω, ω)and Fn1−1 (ω, ω).
if , ip ≤ i
Proof. Given an independent family I ⊆ [ω]ω with ℵ0 ≤ |I| < if .
Convert I to an independent family of functions C ⊆ ω ω with the same
size as I. We get a witness h for the nonmaximality of C, and thenconvert h to a set H which is a witness for the nonmaximality of I.
Similarly for ip .

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Cardinals associated with functions and permutations

d ≤ if
Proof. Given an independent family I ⊆ ω ω such that ℵ1 ≤ |I| < d.
Take a model M of (a large fragment of) ZFC with |M| = |I| and I ∈ M.
Construct a sequence {nk : k < ω} ⊆ ω with n0 = 0 so thatfor any g ∈ M ∩ ω ω there are infinitely many k such that g(nk ) < nk+1 .
(This part uses the assumption |M| < d.)Let {fk : k < ω} ⊆ I be a sequence in M without repetitions. Define∪
h=fk [nk , nk+1 ).
k<ωThen h is a witness for the nonmaximality of I.
(This part uses the assumption that the sequence of fk ’s lies in M.)

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Cardinals associated with functions and permutations

d ≤ if
Proof. Given an independent family I ⊆ ω ω such that ℵ1 ≤ |I| < d.
Take a model M of (a large fragment of) ZFC with |M| = |I| and I ∈ M.
Construct a sequence {nk : k < ω} ⊆ ω with n0 = 0 so thatfor any g ∈ M ∩ ω ω there are infinitely many k such that g(nk ) < nk+1 .
(This part uses the assumption |M| < d.)Let {fk : k < ω} ⊆ I be a sequence in M without repetitions. Define∪h=fk [nk , nk+1 ).
k<ωThen h is a witness for the nonmaximality of I.
(This part uses the assumption that the sequence of fk ’s lies in M.)

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Cardinals associated with functions and permutations

d ≤ if
Proof. Given an independent family I ⊆ ω ω such that ℵ1 ≤ |I| < d.
Take a model M of (a large fragment of) ZFC with |M| = |I| and I ∈ M.
Construct a sequence {nk : k < ω} ⊆ ω with n0 = 0 so thatfor any g ∈ M ∩ ω ω there are infinitely many k such that g(nk ) < nk+1 .
(This part uses the assumption |M| < d.)Let {fk : k < ω} ⊆ I be a sequence in M without repetitions. Define∪h=fk [nk , nk+1 ).
k<ωThen h is a witness for the nonmaximality of I.
(This part uses the assumption that the sequence of fk ’s lies in M.)

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Cardinals associated with functions and permutations

ciae

ap

aif

sf = non(M) = sprprbipsdcov(M) = rfp
ℵ1

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Cardinals associated with functions and permutations

Open Problems

1Is rp = cov(M)?

2

Does d ≤ ip hold?

3

Is each of the following statements consistent with ZFC?
cov(M) < ip
if < iip < i4

Are any strict inequalities between if and ip consistent with ZFC?
(Any possible strict inequalities between ae and ap is still unknown.)

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Cardinals associated with functions and permutations

References
1

Bartoszyński, T.: Combinatorial aspects of measure and category,Fundamenta Mathematicae 127(3), 225–239 (1987).

2

Bell, M.G..: On the combinatorial principle P(c), Fundamenta
Mathematicae 114, 149–157 (1981).

3Blass, A.: Combinatorial Cardinal Characteristics of the Continuum,
Handbook of set theory (M. Foreman and A. Kanamori, editors), vol.
1, Springer, 395–490 (2010).

4Blass, A., Hyttinen, T., Zhang, Y.: Mad families and their neighbors,
preprint, (2007).

5

Brendle, J., Spinas, O., Zhang, Y.: Uniformity of the Meager Idealand Maximal Cofinitary Groups, Journal of Algebra 232, 209–225(2000).

6Enderton, H.B.: A Mathematical Introduction to Logic, Academic
Press (2001).
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7Fremlin, D.H., Shelah, S.: On partitions of the real line, Israel
Journal of Mathematics 32, 299–304 (1979).

8Halbeisen, L.J.: Combinatorial Set Theory: With a Gentle
Introduction to Forcing, Springer, 2017.

9Kunen, K.: Set Theory, Studies in Logic: Mathematical Logic andFoundations, Vol. 34, Colledge Publication, (2011).

10Shelah, S.: Remarks on some cardinal invariants of the continuum, inOpen Problems in Topology (J. van Mill and G. Reed, eds.),
North-Holland, Amsterdam, 217–218 (1990).

11Sonpanow, N., Vejjajiva, P.: Independent families of functions andpermutations, Mathematical Logic Quarterly 66, 311–315 (2020).

12Zhang, Y.: On a Class of M.A.D. Families, The Journal of Symbolic
Logic 64(2), 737–746 (1999).

13Zhang, Y.: Adjoining almost disjoint permutations, Mathematical
Logic Quarterly 48, 189–193 (2002).
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THANK YOU

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Cardinals associated with functions and permutations