Partition Relation Perspectives

Jean A. Larson

The Role of the Hiigher Infinite in Mathematics and Other Disciplines Newton Institute, Cambridge England, 12/15/15 Mirna’s doctoral work In 1993, Mirna Dzamonja received her PhD under the direction of Ken Kunen at the University of Wisconsin, Madison. Mirna’s early career

I From 1993-1995, Mirna was a postdoctoral fellow at the Hebrew University of Jerusalem.

I From 1995-1998, she was a visiting assistant professor at the University of Wisconsin, Madison.

I I first met her at an Annual Association of Symbolic Logic Meeting held in Madison in March 1996. University of East Anglia UEA

Mirna has been at the University of East Anglia since 1998. Mirna’s students Mirna’s students

I Katherine Thompson, University of East Anglia (UEA) 2003. I Grace Piper, UEA, 2003 I Firoz Shaikh, UEA, 2007 I Alexander Primavesi, UEA, 2011 I Omar Selim, 2012 I Gregor Dolinar, University of Ljubljana, 2013 I Sharifa Al Mahrouqi, UEA, 2013 Her co-authors Uri Abraham Lorenz Halbeisen Justin Tatch Moore Arthur Apter Joel David Hamkins Charles Morgan Taras Banakh Michael Hruˇs´ak Eva Murtinov´a Tomek Bartoszy´nski Istv´anJuh´asz Grzegorz Plebanek Robert Bonnet P´eterKomj´ath Anatoli Plichko Piotr Borodulin-Nadzieja Ken Kunen Saharon Shelah James Cummings Jean Larson Katherine Thompson Gregor Dolinar William Mitchell Jouko V¨a¨an¨anen Sy-David Friedman Source: MathSciNet Balanced BHT Theorem

Balanced Baumgartner-Hajnal-Todorcevic (BHT) Theorem, 1993 For every regular uncountable κ, ξ < κ, and k < ω,

κ + 2 (2 ) → (κ + ξ)k .

This theorem has a very nice proof using ideals that come from chains of elementary submodels. Balanced BHT Theorem

Ari Brodsky’s Balanced BHT Theorem for Trees, 2014 For every infinite regular cardinal κ, ξ with 2|ξ| < κ, and k < ω,

<κ
2 non- 2 -special tree → (κ + ξ)k .

Brodsky developed a theory of stationary subsets of trees in order to prove this theorem and also used ideals that come from chains of elementary submodels. Polarized Partitions and Cardinal Invariants

Theorem (Paul Erd˝osand Richard Rado, 1956)

 ℵ   ℵ 1,1 The Continuum Hypothesis implies 1 1 ω 9 ω 2

 θ   θ 1,1 Here → means for every function c : θ × ω → 2 ω ω 2 there are A ⊆ θ and B ⊆ ω such that |A| = θ, B is infinite and c  (A × B) is constant. Polarized Partitions and Cardinal Invariants

Theorem (Shimon Garti and Saharon Shelah, 2014) ω It is consistent with ZFC, that all cardinals θ with ℵ1 ≤ θ ≤ 2 ,

 θ   θ 1,1 → ω ω 2 Earlier they had shown the partition relation was true for θ < s (splitting number) and here show it is true for r < θ (reaping number). Their model has ℵ1 = r =< s = ℵ2 = c. Question: is it consistent that r < s and c > ℵ2. Topological reduction of colors

Claribet Pi˜nahas investigated for given countable ordinals α and beta, the minimum number of colors m needed, so that if pairs from α are colored with finitely many colors (at least 2), then there is a topological copy of beta whose pairs realize at most m colors.

2 This may be expressed compactly as (∀` > 1)(α →top (β)`,m). Theorem (Pi˜na,2013, 2015)

ω·n 2 2 I for all n > 1, (∀` > 1)(ω →top (ω + 1)`,6), but ω·n 2 2 (ω 9top (ω + 1)9,5). ωk 1 2 I for all k > 1, (∀` > 1)(ω →top (ω )`,6), but ωk 2 2 ω 9top (ω + 1)3(k+1),5. Topological Pigeonhole Principle

Building on the ordinal pigeonhold principle of Eric Milner and ord Richard Rado which calculates P (αi ), and work of Bill Weiss, of Prikry and Solovay, and of Shelah, Jacob Hilton constructed a pigeonhole principle for ordinal topological spaces. Here is the theorem he describes as his main breakthrough. Theorem (Jacob Hilton, JSL, to appear)

Suppose 0 < α1, α2, . . . , αk < ω1. 1. Ptop(ωα1 + 1, ωα2 + 1, . . . , ωαk + 1) = ωα1#α2#...#αk + 1. ord 2. Ptop(ωα1 , ωα2 , . . . , ωαk ) = ωP (α,α2,...,αk ).

a ap a ap If α = ω 0 · m0 + ··· + ω · mp and β = ω 0 · n0 + ··· + ω · np where a0 > a1 > ··· > ap and m0, m1,... mp, n0, n1,... np < ω, then the Hessenberg sum (natural sum) is defined by a ap α#β = ω 0 · (m0 + n0) + ··· + ω · (mp + np) Topological Pigeonhold Principle

There is a reason why powers of ω and their successors appear so often. We say X and Y are biembeddable and write X ≡ Y if X is homeomorphic to a subspace of Y and Y is homeomorphic to a subspace of X . Lemma (Hilton) n If Y ≡ V and Xi ≡ Ui for all i < κ, then Y →top (Xi )i<κ if and n only if V →top (Ui )i<κ. Topological Ramsey numbers

In their search for Ramsey numbers of ordinal topological spaces, Caicedo and Hilton looked at two kinds of goals: Rtop when the goals were homeomorphic to given ordinal spaces and Rcl when the goals that were both order-isomorphic to given ordinal spaces and closed in their own supremum. top As noted above, Hilton proved a pigeonhole principle P (αi )i<κ for goals that were topological spaces and Caicedo and Hilton found many cases of Hilton’s proof carried over to the closed cl pigeonhole principle, P (αi )i<κ, and they computed the values in the remaining cases quite handily. Topological Ramsey numbers

top They write R (αi )i<κ for the topological Ramsey number, namely the least ordinal β such that

2 β →top (αi )i<κ

cl Similarly, they write R (αi )i<κ for the closed Ramsey number, namely the least ordinal β such that

2 β →cl (αi )i<κ Topological Ramsey numbers

Paul Erd˝osand Eric Milner proved the following theorem for ordinal partition relations:

Theorem (Erd˝os, Milner) If ωα → (ω1+β, k)2, then ωα+β → (ω1+β, 2k)2.

Andres Caicedo and Jacob Hilton found a counterpart which they call the Weak topological Erd˝os-MilnerTheorem.

Theorem (Caicedo, Hilton) Let α and β be countable non-zero ordinals, and let k > 1 be a positive integer. If ωα β 2 ω →top (ω , k) , then ωα·β β 2 ω →top (ω , k + 1) , Topological Ramsey numbers

In a corollary to their main theorem, Caicedo and Hilton prove these Ramsey numbers coincide in certain cases and give upper bounds for them. Theorem (Caicedo, Hilton)

Suppose 0 < α < ω1 and k, m and n are positive integers. 1. Rtop(α, k) is countable. 2. Rtop(ωωα , k + 1) = Rcl(ωωα , k + 1) ≤ ωωα·k . 3. Rtop(ωωα + 1, k + 1) = Rcl(ωωα + 1, k + 1) and the common value is bounded by ωωα·k + 1 if α is infinite, and bounded by ωω(n+1)·k−1 + 1 if α = n is finite. 4. Rtop(ωn ·m, k +2) = Rcl(ωn ·m, k +2) ≤ ωωk ·n ·R(m, k +2)+1. Topological Ramsey Theory

Question: is there a topological partition ordinal α > ω? 2 That is, is there α > ω for which α →top (α, 3) ?

Question: is it possible to reduce the computation of Rtop(α, k) to finite combinatorial problems, even for α < ω2? Halpern-L¨auchliTheorem

Copies of illustrations of a k-dense subset and a k-x-dense subset from Ramsey Theory by Todorcevic:

Q Q A k-dense matrix (k-~x-dense matrix) for i

(Asymmetric Version of the Halpern-L¨auchliTheorem) For every coloring of the finite product of finitely branching trees, Q i

Laver’s Conjecture: For every (level set) coloring of the infinite product of downwards Q closed perfect i<ω Ti = K0 ∪ K1, there is some j < 2 and some Q ~x ∈ i<ω Ti such that for each k < ω, Kj contains a k-~x-dense (level set) matrix.

Joint work with Dˇzamonjaand Mitchell

Theorem (Denis Devlin, 1979)

n n Q → (Q)<ω,tn and Q 9 (Q)<ω,tn−1,

where tn is the nth tangent number.

For regular cardinals κ, there is a natural order ≤Q extending κ> κ> lexicographic order on 2 such that Qκ = h 2, ≤Q i is a κ-dense linear order. Joint work with Dˇzamonjaand Mitchell

Theorem (Dˇzamonja, Larson, Mitchell, 2009) + For every positive integer m there is tm < ω such that for any cardinal κ which is measurable after generically adding λ many 2m Cohen subsets of κ where λ → (κ)2κ , the κ-dense linear order Qκ satisfies

m m κ → ( )κ) + and κ ( )κ) + , Q Q <ω,tm Q 9 Q <ω,tm −1

+ where tm is the cardinality of a finite set of trees.

+ + Furthermore, for m ≥ 3, tm > tm: t3 = 16 < 20 = t3 ; + + t4 = 272 < 776 = t4 ; t5 = 7936 < 151, 184 = t5 . Joint work with Dˇzamonjaand Mitchell

A key tool:

Theorem (Saharon Shelah, 1988) Suppose that m < ω and κ is a cardinal which is measurable in the generic extension obtained by adding λ Cohen subsets of κ where 2m λ → (κ)2κ . Then for any coloring d of the m-element antichains of κ>2 into σ < κ colors, and any well-ordering ≺ of the levels of κ>2, there is a strong embedding e : κ>2 → κ>2 and a dense set of elements w such that

I e(s) ≺ e(t) for all s ≺ t from Cone(w), and

I d(e[a]) = d(e[b]) for all ≺-similar m-element antichains a and b of Cone(w). Work with Dˇzamonjaand Mitchell

Based on work by Hajnal and Komj´ath,the positive partition relation for Qκ does not follow from any large cardinal hypothesis on κ.

Theorem (Andr´asHajnal and P´eterKomj´ath)

There is a forcing of size ℵ1 which adds an order type θ of size ℵ1 2 with the property that ψ 9 [θ]ω1 for every order type ψ, regardless of its size. Work with Dˇzamonjaand Mitchell

Our proof uses the theorem of Shelah quoted above to show that all the types we say occur do occur in every large set. m Does Qκ 9 (Q) + hold for sufficiently large m < ω for some <κ,tm−1 regular cardinal which is not measureable after generically adding λ 2m Cohen subsets of κ where λ → (κ)2κ ? Remembering Mary Ellen Rudin Bibliography

I J. Baumgartner, A. Hajnal, and S. Todorcevic. Extensions of the Erd˝os-Radotheorem. In Finite and Infinite Combinatorics in Sets and Logic (Banff, AB, 1991), pages 1–17. Kluwer Acad. Publ., Dordrecht, 1993.

I J. Baumgartner. Partition relations for countable topological spaces. J. Combin. Theory Ser. A, 43(2):178–195, 1986.

I A. Brodsky. A theory of stationary trees and the balanced Baumgartner-Hajnal-Todorcevic theorem for trees. Acta Math. Hugar., 144(2):285-352.

I A. Caicedo and J. Hilton. Topological Ramsey numbers and countable ordinals. preprint, September 30, 2015.

I Denis Devlin. Some partition theorems and ultrafilters on ω. Ph.D. thesis, Dartmouth College, 1979. Bibilography

I M. Dzamonja, J. Larson, and W. Mitchell. A partition theorem for a large dense linear order. Israel J. Math 171:237-284, 2009.

I P. Erd˝osand R. Rado. A partition calculus in set theory. Bull. Amer. Math. Soc., 62:427-489, 1956.

I A. Hajnal and P. Komj´ath.A strongly non-Ramsey order type. Combinatorica, 17:363-367, 1997.

I J. D. Halpern and H. L¨auchli.A partition theorem. Trans. Amer. Math. Soc., 124:36–367, 1966.

I J. Hilton. The topological pegeonhole principle for ordinals. J. Symbolic Logic, to apppear (also on arXiv). Bibilography

I R. Laver. Products of infinitely many perfect trees. J. London Math. Soc. (2), 29(3):385–396, 1984.

I S. Garti and S. Saharon. Partition calculus and cardinal invariants. J. Math. Soc. Japan, 66:425-434, 2014.

I E. C. Milner and R. Rado, The pigeon-hole principle for ordinal numbers, Proc. of the AMS, 3(1):750-768, 1965.

I S. Shelah. Was Sierpi´nskiright? I. Israel J. Math., 62:335–380, 1988. Sh: 276.

I S. Todorcevic. Ramsey Theory. Annals of Mathematics Studies, 174. Prinction University Press, Princeton, NJ, 2010. viii+287:2010. Thank you!