Disjoint refinements
Menachem Kojman
Ben-Gurion University, Beer Sheva
July 24, 2013
Menachem Kojman
Disjoint refinements
Intersections conditions
Let F be a family of infinite sets; if every A ∈ F has
cardinality |A| = ρ, then
S F is ρ-uniform; the universe of F is
the set V = V (F) = F.
Convention
F is always a ρ-uniform family of sets, ρ ≥ ℵ0 and λ = |F|.
F satisfies the intersection condition C (θ, κ) if the
intersection of any θ members from F has cardinality < κ.
Example: C (2, 1) means the family is pairwise disjoint;
F is is almost disjoint if C (2, ρ) holds; it is ν-almost disjoint,
if C (2, ν) holds.
Menachem Kojman
Disjoint refinements
Splitting conditions
A disjoint refinement of F = {Aα : α < λ} is a ρ-uniform
pairwise disjoint {dα : α < λ} such that dα ⊆ Aα for all
α < λ. Let DR(F) stand for “F has a disjoint refinement.”
A stronger property than having a disjoint refinement is
essential disjointness. F is essentially disjoint if for each
α < λ there exists cα ∈ [Aα ]<ρ such that {Aα \ cα : α < λ} is
pairwise disjoint. Let ED(F) stand for “F is essentially
disjoint”.
A weaker property than having a disjoint refinement is having
an injective choice function and an even weaker one, by a
theorem of Komjáth and Hoffmann, is Property B (after F.
Bernstein): there is a set B such that ∅ =
6 Aα ∩ B 6= Aα for all
α < λ. Let B(F) stand for “F has property B”.
Menachem Kojman
Disjoint refinements
Miller theorems
Theorem
For every n < ω and ρ ≥ ℵ0 , for every ρ-uniform family F,
1
Miller, 1937: C (ρ+ , n) =⇒ B(F).
2
Komjáth, 1984: C (ρ+ , n) =⇒ DR(F).
3
Komjáth, 1984: C (2, n) =⇒ ED(F).
Menachem Kojman
Disjoint refinements
Proofs
The proofs go by induction on λ. If λ ≤ ρ then DR(F) is simply
true (Sierpiński, early 1900) and the implication
C (2, n) =⇒ ER(F) is obvious. So assume λ > ρ.
Since C (ρ+ , n) holds, every X ∈ [V ]n can be a subset of at most ρ
members of F. By Skolem-Löwenheim, V = V (F) has a filtration
{Vα : α < λ} in which Vα is n-closed: if X ∈ [Vα ]n and
X ⊆ A ∈ F then A ⊆ Vα or, equivalently,
A 6⊆ Vα =⇒ |A ∩ Vα | < n.
Now DR(Vα+1 \ Vα ) holds by the induction hypothesis, with, say
dA ∈ [A]ρ witnessing this. For A ∈ Vα+1 \ Vα let dA0 = da \ Vα and
now {dA0 : A ∈ F} is a disjoint refinement. Similarly, if cA ∈ [A]<ρ
witness essential disjointness, replace cA with cA ∪ (A ∩ Vα ).
Menachem Kojman
Disjoint refinements
Relaxing C (ρ+ , n) to C (ρ+ , ν)
The condition C (ρ+ , n) is too strong for large ρ. Can it be relaxed
to C (ρ+ , ν) with an infinite ν? For a given ν, can we find µ such
that C (ρ+ , ν) suffices for all ρ ≥ µ in Miller theorems?
The problem is that there is no Skolem-Löwenheim theorem for
ν-ary operations: closing under such operations may increase
cardinality, and an increasing union of closed sets may fail to be
closed. Yet, in this context, the closure operator is
anti-monotone, which makes it behave well at non-critical limits.
Menachem Kojman
Disjoint refinements
With additional axioms
Theorem (Erdős and Hajnal, 1961)
Assuming the GCH, for every infinite cardinal ν and ρ > ν + , for
every ρ-uniform F,
C (ρ+ , ν) =⇒ B(F).
Really, the same proof gives DR(F).
Theorem (Hajnal, Juhász and Shelah, 1986)
If for every µ > 2ν there are fewer than ν + cardinals in the interval
(µ, µν ), then for every ρ > (2ν )+ , for every ρ-uniform F,
C (2, ν) =⇒ ED(F).
Menachem Kojman
Disjoint refinements
In ZFC
Theorem (K. 2012)
For every infinite cardinal ν and ρ ≥ iω (ν), for every ρ-uniform F
1
C (ρ+ , ν) =⇒ DR(F);
2
C (2, ν) =⇒ ED(F).
The second part was proved by Soukup for ρ = iω .
Menachem Kojman
Disjoint refinements
Density
Definition
A family D ⊆ [λ]ν is dense in [λ]θ if
(∀Y ∈ [λ]θ )(∃X ∈ D) X ⊆ Y .
The (ν, θ)-density of λ, denoted D(λ, ν, θ), is the least
cardinality of D ⊆ [λ]ν which is dense in [λ]θ .
D(λ, ν) = D(λ, ν, ν).
Menachem Kojman
Disjoint refinements
The Hajnal-Juhász-Shelah Theorem
Lemma
D(λ, ν) = λ for all λ ≥ 2ν if for all µ > 2ν there are no cardinals
of cofinality cf ν in the interval (µ, µν ).
Proof.
By induction. Obvious for λ = 2ν and every other power of ν, µν ;
persists at limits of cofinality different than cf ν. By the
assumption, these cases are enough.
With density you get closed sets. For critical cofinalities pay
another cardinal.
Menachem Kojman
Disjoint refinements
In ZFC
Theorem
For every cardinal ν, for every λ ≥ iω (ν), for almost all n < ω,
D(λ, in (ν)) = λ.
This is an easy consequence of Shelah’s revised GCH theorem.
Menachem Kojman
Disjoint refinements
Komjáths comparison theorem
We say that two ρ-uniform families F, G are similar if they can be
enumerated F = {Aα : α < λ}, G = {Bα : α < λ} such that
|Aα ∩ Aβ | = |Bα ∩ Bβ | for all α < β < λ.
Property B, the existence of an injective choice function and the
existence of a disjoint refinement are not invariant under similarity.
Theorem (Komjáth 1986)
If F and G are similar ℵ0 -uniform almost disjoint families and F is
essentially disjoint, then also G is essentially disjoint. Similarity can
be relaxed to |Aα ∩ Aβ | ≥ |Bα ∩ Bβ |.
Menachem Kojman
Disjoint refinements
Theorem (K. 2012)
Suppose µ is a strong limit cardinal and ρ ≥ µ. Suppose
F = {Aα : α < λ}, G = {Bα : α < λ} are ρ-uniform and for every
α < β < λ it holds that
|Aα ∩ Aβ | ≥ |Bα ∩ Bβ |
Then if F is < µ-essentially disjoint, so is G, provided that every
subfamily of G of cardinality ρ is µ-e.d.
Conflict-free colorings
A coloring c : V → κ is conflict-free if for each A ∈ F there is a
point x ∈ A such that c(x) 6= c(y ) for all y ∈ A such that y 6= x.
Theorem (Soukup 2012)
If ν < iω ≤ λ, then every F ⊆ [λ]≥iω that satisfies C (2, ν) has a
conflict-free coloring with iω colors.
Theorem (K. 2012)
For every cardinal ν and ρ ≥ iω (ν), every ρ-uniform family F that
satisfies C (ρ+ , ν) has a conflict-free coloring with ρ+ colors.
This follows from yet another strengthening of Miller’s theorem.
Menachem Kojman
Disjoint refinements
A general scheme
Finite Exponent
Infinite Density
λin (k) = λ
←→
D(λ, in (κ)) = λ
For all n ∈ N
←→
For almost all n ∈ N
λ ≥ ℵ0
←→
λ ≥ iω (κ)
(ℵ0 =) iω (k)
←→
iω (κ)
k ∈N
←→
κ a cardinal
Menachem Kojman
Disjoint refinements
Lower bounds?
To show that i2 (ν) is not enough in place of iω (ν) for all
sufficiently large ν, the singular cardinals hypothesis has to violated
in a bad way. By Gitik-Magidor methods, from a class of large
cardinals there are models in which the gaps between µ and µcf µ
are unbounded.
A version of Shelah’s Weak Hypothesis is:
SWH(n)
There is no in (ν) < ρ < iω (ν) and a family F ⊆ [ρ]ρ of
cardinality |F| ≥ ρ+ which satisfies C (2, ν).
Proving the consistency of this will prove that in (ν) is not enough.
Menachem Kojman
Disjoint refinements