Intersecting families and definability - Andrés E. Caicedo · Intersecting families and...

Intersecting families and definability

Andres E. Caicedo

Department of MathematicsBoise State University

September 19, 2008

Caicedo Intersecting families and definability

IntroductionGlimm-Effros reductions

Intersecting families

Introduction

I would like to present some results and problems that combinefinite combinatorics and mathematical logic.This is joint work with John Clemens, Clinton Conley, andBenjamin Miller.




Introduction

I would like to present some results and problems that combinefinite combinatorics and mathematical logic.This is joint work with John Clemens, Clinton Conley, andBenjamin Miller.




These results were motivated by research in Descriptive SetTheory. The basic fact from which the results follow wasneeded in order to establish a Glimm-Effros dichotomy.The results we obtained have also found applications in theDescriptive Set theoretic study of percolation theory.




These results were motivated by research in Descriptive SetTheory. The basic fact from which the results follow wasneeded in order to establish a Glimm-Effros dichotomy.The results we obtained have also found applications in theDescriptive Set theoretic study of percolation theory.




Glimm-Effros reductions

I will concentrate on the combinatorial content of the resultsrather than their applications, but I want to start with a fewwords about the work that led to this research.A Polish space is a separable complete metrizable space.Typical examples include R, the Cantor set, and Hilbert space.A reduction from an equivalence relation E on a space X toanother equivalence relation F on some other space Y is amap π : X → Y such that

xEy ⇐⇒ π(x)Fπ(y)

for any x , y ∈ X .


















The relation E is called countable iff each class is countable.When E and F are countable, X and Y are Polish, and thegraphs of E , F , and π are Borel (as subsets of X × X , Y × Yand X × Y , respectively), we write E ≤B F .Many natural classification problems in mathematics can beformalized as questions about reductions between countableBorel equivalence relations on Polish spaces, so it is natural tostudy the quasi-order ≤B, and Descriptive Set Theory has beenactively involved in this pursuit over the last two decades.




The relation E is called countable iff each class is countable.When E and F are countable, X and Y are Polish, and thegraphs of E , F , and π are Borel (as subsets of X × X , Y × Yand X × Y , respectively), we write E ≤B F .Many natural classification problems in mathematics can beformalized as questions about reductions between countableBorel equivalence relations on Polish spaces, so it is natural tostudy the quasi-order ≤B, and Descriptive Set Theory has beenactively involved in this pursuit over the last two decades.




The relations that are ≤B-minimal are called smooth. Theirclassification problem is as simple as possible. Examples are=R (equality of real numbers), and the problem of identifyingwhen two linear transformations of Rn into itself are equivalentup to change of bases.A significant result, originated in work of Glimm-Effros onoperator algebras in the 1960s, is that there are ≤B-minimalnon-smooth relations. The canonical example is E0 on {0,1}Ndefined by

xE0y ⇐⇒ ∃n ∀m ≥ n (x(m) = y(m)).




The relations that are ≤B-minimal are called smooth. Theirclassification problem is as simple as possible. Examples are=R (equality of real numbers), and the problem of identifyingwhen two linear transformations of Rn into itself are equivalentup to change of bases.A significant result, originated in work of Glimm-Effros onoperator algebras in the 1960s, is that there are ≤B-minimalnon-smooth relations. The canonical example is E0 on {0,1}Ndefined by

xE0y ⇐⇒ ∃n ∀m ≥ n (x(m) = y(m)).




Very roughly, this result is proved as follows: Given a countableBorel equivalence relation E , there is a natural attempt atrecursively building a continuous injective reduction of E0 intoE , which essentially entails trying to build up copies oflevel-by-level approximations to E0 within E . If this attempt failsto produce the desired reduction, then it provides us with a(Borel) way of tagging E-classes with real numbers, thusshowing that E is smooth.




This last part of the proof (the “tagging” argument) factorsthrough the study of certain σ-ideal IE . Clemens, Conley, andMiller worked on clarifying the role that IE plays in determiningthe place of E within the quasi-order ≤B.Lying at the heart of their result is the fact that wheneverΦ : X → Y is Borel, then either there is a Borel perturbation ofΦ which is a homomorphism from IE to IF , or else there is acontinuous injective reduction π of E0 into E with the propertythat the points of the form Φ ◦ π(x), for x ∈ {0,1}N, are pairwiseF -inequivalent.




This last part of the proof (the “tagging” argument) factorsthrough the study of certain σ-ideal IE . Clemens, Conley, andMiller worked on clarifying the role that IE plays in determiningthe place of E within the quasi-order ≤B.Lying at the heart of their result is the fact that wheneverΦ : X → Y is Borel, then either there is a Borel perturbation ofΦ which is a homomorphism from IE to IF , or else there is acontinuous injective reduction π of E0 into E with the propertythat the points of the form Φ ◦ π(x), for x ∈ {0,1}N, are pairwiseF -inequivalent.




For this, again, there is a natural attempt at recursively buildingthe desired reduction of E0 to E . Failure of this attempt can betranslated into an assignment of a (possibly infinite) intersectingfamily of finite sets to certain Borel sets. In this case, one thenobtains the desired Borel perturbation of Φ by appealing to thecombinatorial results which I now proceed to present.




DefinabilityUpper boundsLow familiesMinimal families3-minimal familiesSome questions


A family A of finite subsets of a set X is intersecting iffwhenever A,B ∈ A then A ∩ B 6= ∅.Both X and A could be infinite. It was somewhat surprising torealize that, nevertheless, whenever A is intersecting, one candefine from A a finite subset of X .For example, letting [X ]n denote the collection of n-elementsubsets of X , suppose A ⊆ [X ]2 is intersecting. Then either|A| ≤ 3 or else there is a (unique) x such that x ∈ A for allA ∈ A.We find quantitative versions of this observation.























Definability

We begin by formalizing the notion of definability we use.

DefinitionGiven a set X and A ⊆ P(X ), let

AX = (X t P(X ),A, ∈),

where t denotes disjoint union, and

α ∈β ⇐⇒ α ∈ X , β ⊆ X , and α ∈ β.

We abuse notation and write ∪,∈. Working with the structureAX stops the internal set structure of the elements of X fromallowing us to define any objects.





Definability



AX = (X t P(X ),A, ∈),


α ∈β ⇐⇒ α ∈ X , β ⊆ X , and α ∈ β.






Definability



AX = (X t P(X ),A, ∈),


α ∈β ⇐⇒ α ∈ X , β ⊆ X , and α ∈ β.






We say that Y ⊆ X ∪P(X ) is definable from A iff Y is first-orderdefinable without parameters in AX . This formal notioncaptures our intuitive understanding of what being definablemeans in this context.

For example,1 For all n, [X ]n is definable from A.2⋃A is definable from A.

3 The sets definable from A are closed under (finite)intersections, unions, and complements.

































Upper bounds

TheoremLet n be a positive integer.

1 If A ⊆ [X ]≤n is intersecting, then there is a non-emptyintersecting A′ ⊆ [X ]≤n definable from A, and of size atmost nn − n + 1.

2 If A ⊆ [X ]≤n is intersecting, then there is a non-emptysubset of X definable from A, and of size at most n2 − 1.





Upper bounds








Upper bounds








Here is the proof: For each A ⊆ P(X ) and D ⊆ X , let

degA(D) = |{A ∈ A : D ⊆ A}|.

An A-extension (or, simply, extension) of D is any A ∈ A withD ⊆ A. For each non-empty intersecting A ⊆ [X ]n and eachm ≤ n, let

dm = supD∈[X ]m

degA(D).

In particular, dn = 1, but some dm could be infinite. Let

Am = {D ∈ [X ]m : degA(D) > ndm+1}.

Notice that Am is definable from A, as long as dm+1 is finite.






degA(D) = |{A ∈ A : D ⊆ A}|.


dm = supD∈[X ]m

degA(D).


Am = {D ∈ [X ]m : degA(D) > ndm+1}.







degA(D) = |{A ∈ A : D ⊆ A}|.


dm = supD∈[X ]m

degA(D).


Am = {D ∈ [X ]m : degA(D) > ndm+1}.







degA(D) = |{A ∈ A : D ⊆ A}|.


dm = supD∈[X ]m

degA(D).


Am = {D ∈ [X ]m : degA(D) > ndm+1}.






We claim that if A ⊆ [X ]n is intersecting, then so is each Am(m < n).Otherwise, there are disjoint D0,D1 ∈ Am. We obtain acontradiction by finding disjoint A0,A1 ∈ A such that Ai is anextension of Di , i = 0,1.To do this, notice that for each x ∈ X \ D0, at most dm+1extensions of D0 contain x , or else degA(D0 ∪ {x}) > dm+1.Consequently, no more than mdm+1 extensions of D0 can meetD1, and (by definition of Am) we can find A0 ∈ A extending D0and disjoint from D1.Similarly, no more than ndm+1 extensions of D1 can meet A0,so there is an A1 ∈ A extending D1 and disjoint from A0,contradiction.




















It follows that each Am, m < n, is intersecting. However, it ispossible that some of the Am are actually empty.Suppose now that Am 6= ∅ and that m is largest with thisproperty. By induction on n −m it follows that each di ,m < i ≤ n, is finite. In particular, dm+1 is finite, and thereforeAm is definable from A.





It follows that each Am, m < n, is intersecting. However, it ispossible that some of the Am are actually empty.Suppose now that Am 6= ∅ and that m is largest with thisproperty. By induction on n −m it follows that each di ,m < i ≤ n, is finite. In particular, dm+1 is finite, and thereforeAm is definable from A.





Now we proceed by induction on n to show that if A ⊆ [X ]n isintersecting, then there exists a non-empty intersecting familyA′ ⊆ [X ]≤n definable from A with |A′| ≤ nn − n + 1.The result is clear when n = 1 (with A′ = A). If for some m < n,Am is non-empty, definable, and intersecting, then theconclusion follows from the inductive hypothesis.Thus, by the argument just shown, we may assume that for allm < n, Am is empty. That is, for all m < n, dm ≤ ndm+1 and, inparticular, d1 ≤ nn−1. Fix any A ∈ A. Every point in A iscontained in at most nn−1 − 1 additional sets in A. Since everyset in A meets A, we have

|A| ≤ n(nn−1 − 1) + 1 = nn − n + 1.






|A| ≤ n(nn−1 − 1) + 1 = nn − n + 1.






|A| ≤ n(nn−1 − 1) + 1 = nn − n + 1.





It follows that from any intersecting A ⊆ [X ]≤n we can define asubset of X of size at most n(nn−n + 1) ≈ nn+1. We now arguethat actually we can define a set of size strictly less than n2.For we can suppose that A ⊆ [X ]n is finite. Then d1 is finite. Let

Y = {x ∈ X : degA(x) = d1}.

Then 1n d1|Y | ≤ 1

n∑

x∈X degA(x) = |A| ≤ 1 + n(d1 − 1), thus

|Y | ≤ n2 − nd1

(n − 1).

This completes the proof.






Y = {x ∈ X : degA(x) = d1}.

Then 1n d1|Y | ≤ 1

n∑

x∈X degA(x) = |A| ≤ 1 + n(d1 − 1), thus

|Y | ≤ n2 − nd1

(n − 1).







Y = {x ∈ X : degA(x) = d1}.

Then 1n d1|Y | ≤ 1

n∑

x∈X degA(x) = |A| ≤ 1 + n(d1 − 1), thus

|Y | ≤ n2 − nd1

(n − 1).







Y = {x ∈ X : degA(x) = d1}.

Then 1n d1|Y | ≤ 1

n∑

x∈X degA(x) = |A| ≤ 1 + n(d1 − 1), thus

|Y | ≤ n2 − nd1

(n − 1).






Low families

To obtain lower bounds, we need a condition ensuringnon-definability.

LemmaAssume X is finite and A ⊆ P(X ). Let B ⊆ X ∪P(X ). Then B isdefinable from A iff B is invariant under Aut(AX ).

Definition

A ⊆ P(X ) is low iff Aut(AX ) acts transitively on both X and A.

In view of the lemma above, for finite X this is equivalent tosaying that A is low iff the only subsets of X definable from Aare X and ∅.





Low families



Definition







Low families



Definition







Low families



Definition







Theorem1 For all n, there is a set X of cardinality n(n + 1)/2 and a

low intersecting family A ⊆ [X ]n.2 If n − 1 is a power of a prime, then there is a set X of

cardinality n2 − n + 1 and a low intersecting familyA ⊆ [X ]n.

This means that the upper bound n2 − 1 found before isessentially best possible.





















How large can a low family be?

DefinitionLet ρ(n) be the largest size of a low intersecting familyA ⊆ [X ]n.

Theorem

ρ(1) = 1, ρ(2) = 3, ρ(3) = 10 and if n > 3, ρ(n) <(n2−2

n−1

).

We expect one should be able to improve this upper boundsignificantly.







Theorem

ρ(1) = 1, ρ(2) = 3, ρ(3) = 10 and if n > 3, ρ(n) <(n2−2

n−1

).








Theorem

ρ(1) = 1, ρ(2) = 3, ρ(3) = 10 and if n > 3, ρ(n) <(n2−2

n−1

).








Theorem

ρ(1) = 1, ρ(2) = 3, ρ(3) = 10 and if n > 3, ρ(n) <(n2−2

n−1

).






Minimal families

Definition

A ⊆ [⋃A]≤n is n-minimal iff A is intersecting and whenever

B ⊆ [⋃

A]≤n is intersecting and definable from A, then|B| ≥ |A|.

Notice that if A is n-minimal, then A ⊆ [⋃A]m for some m ≤ n.

DefinitionLet ψ(n) be the largest size of an n-minimal family.

We expect that if A is n-minimal of size ψ(n) then A ⊆ [⋃A]n,

but this is still open.





Minimal families

Definition


B ⊆ [⋃










Minimal families

Definition


B ⊆ [⋃










Minimal families

Definition


B ⊆ [⋃










We can characterize minimality of A in terms of Aut(AX ), aslong as A is sufficiently large.

TheoremSuppose A ⊆ [X ]n is intersecting and |A| > ψ(n − 1). Then thefollowing are equivalent:

1 A is minimal.2 For all A ∈ [X ]≤n either A /∈ A and there is σ ∈ Aut(AX )

such that A ∩ σ · A = ∅, or else A ∈ [X ]n and|Aut(AX ) · A| ≥ |A|.





























How large is ψ(n)? We already have ψ(n) ≤ nn − n + 1. Thecharacterization in terms of automorphisms allows us to showthe following:

Theorem1 For all n, ψ(n) ≥

(2n−1n

).

2 For all even n > 6, there is an n-minimal A with|⋃A| ≥ n2/4 and |A| > (n/2)n/2.

In addition, the families in this theorem are low, so they alsogive us lower bounds on ρ(n).It is not clear to us which of the two bounds is actually closer toψ(n).







(2n−1n

).









(2n−1n

).









(2n−1n

).









(2n−1n

).







3-minimal families

Let Bn denote the collection of n-minimal families of sets,considered up to isomorphism.For example, B1 = {[1]1} and B2 = {[1]1, [2]2, [3]2}.Given a set of integers A ⊆ n, let An denote the family oftranslations of A taken modulo n.Let L = {A ∈ [6]3 :

∑x∈A x ∈ {1,3,4} (mod 6)}.

Let AOct = {{0,1,2}, {0,4,5}, {1,3,5}, {2,3,4}} ⊆ [6]3.

TheoremB3 ={[1]1, [2]2, [3]3, [4]3,AOct, {0,1,3}5, {0,1,3}6, {0,1,3}7, [5]3,L}.

In particular, ψ(3) = 10.





3-minimal families


∑x∈A x ∈ {1,3,4} (mod 6)}.

Let AOct = {{0,1,2}, {0,4,5}, {1,3,5}, {2,3,4}} ⊆ [6]3.

TheoremB3 ={[1]1, [2]2, [3]3, [4]3,AOct, {0,1,3}5, {0,1,3}6, {0,1,3}7, [5]3,L}.






3-minimal families


∑x∈A x ∈ {1,3,4} (mod 6)}.

Let AOct = {{0,1,2}, {0,4,5}, {1,3,5}, {2,3,4}} ⊆ [6]3.

TheoremB3 ={[1]1, [2]2, [3]3, [4]3,AOct, {0,1,3}5, {0,1,3}6, {0,1,3}7, [5]3,L}.






3-minimal families


∑x∈A x ∈ {1,3,4} (mod 6)}.

Let AOct = {{0,1,2}, {0,4,5}, {1,3,5}, {2,3,4}} ⊆ [6]3.

TheoremB3 ={[1]1, [2]2, [3]3, [4]3,AOct, {0,1,3}5, {0,1,3}6, {0,1,3}7, [5]3,L}.






3-minimal families


∑x∈A x ∈ {1,3,4} (mod 6)}.

Let AOct = {{0,1,2}, {0,4,5}, {1,3,5}, {2,3,4}} ⊆ [6]3.

TheoremB3 ={[1]1, [2]2, [3]3, [4]3,AOct, {0,1,3}5, {0,1,3}6, {0,1,3}7, [5]3,L}.






3-minimal families


∑x∈A x ∈ {1,3,4} (mod 6)}.

Let AOct = {{0,1,2}, {0,4,5}, {1,3,5}, {2,3,4}} ⊆ [6]3.

TheoremB3 ={[1]1, [2]2, [3]3, [4]3,AOct, {0,1,3}5, {0,1,3}6, {0,1,3}7, [5]3,L}.






3-minimal families


∑x∈A x ∈ {1,3,4} (mod 6)}.

Let AOct = {{0,1,2}, {0,4,5}, {1,3,5}, {2,3,4}} ⊆ [6]3.

TheoremB3 ={[1]1, [2]2, [3]3, [4]3,AOct, {0,1,3}5, {0,1,3}6, {0,1,3}7, [5]3,L}.






Some questions

From the results above, ψ(n) < ψ(2n) for all n.

QuestionIs ψ strictly increasing?

All our examples of minimal families are low.

QuestionDoes n-minimality imply lowness?





Some questions









Some questions









Some questions






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