Papers

In Progress

A proof (involving elementary substructures) of a partition relation (with Jean A. Larson)

This article provides a proof of the full Baumgartner-Hajnal-Todorčević theorem (that if P → (ω)¹_ω, then P → (α)²_n for all α < ω₁ and n < ω) that involves a lot of elementary substructures but doesn't involve any forcing or absoluteness. The final version will also present limited generalizations of the result from ω to larger cardinals.

The results presented in this draft were produced about 16 years ago, in the winter and spring of 1999. Jean and I were working together more recently to complete and expand this paper, but we haven't made any progress on it in quite some time now.

Partitioning triples just above a measurable cardinal

This article generalizes some of my earlier work to this new setting. The main result is that if κ is measurable, then κ⁺ → (κ + μ, n)³ for all μ < κ and n < ω. More generally, if κ is measurable, and P → (κ)¹_κ then P → (κ + μ, n)³ for all μ < κ and n < ω. The proof is direct and combinatorial and does not rely on any metamathematical techniques.

I don't think I'll ever get this all written up. Ugh.

A polarized version of the Erdős-Rado theorem

A result of J. Baumgartner and A. Hajnal is extended. The new result is that if λ = 2^<κ, then (λ⁺ : λ⁺⁺) → (λ⁺ : α, (κ : κ + 1)_μ)^1,1 for all α < λ⁺ and μ < cf κ.

This is a simple and pretty result, but I don't know if it's terribly useful.

Published

Even more on partitioning triples of countable ordinals

Proc. Amer. Math. Soc., vol 146, no. 8 (2018), 3529-3539

The new result is that ω₁ → (ω + ω + 1, n)³ for all n < ω. To my knowledge, this is currently the best result on the conjecture of Erdős et al. that ω₁ → (α, n)³ all α < ω₁ and n < ω.

On a result of Szemerédi

J. Symb. Logic, vol. 73, no. 3 (2008), 953-956

This note presents a short proof that if κ is regular with κ ≤ 𝔠, then (ω : κ) → (ω : κ, ω : α)^1,1 for all α < min {𝔭, κ}. This improves somewhat on the (apparently unpublished) result of Szemerédi that Martin's axiom implies that (ω : 𝔠) → (ω : 𝔠, ω : κ)^1,1 for any cardinal κ < 𝔠.

S. Garti and S. Shelah have since produced many beautiful results and compelling open questions in this vein. See, for example, their articles, Strong polarized relations for the continuum, Annals of Combinatorics, vol. 16, pp. 271–276, 2012 (also published as arxiv.org:1103.5195v1 in 2011), and Open and solved problems in infinite combinatorics (published as arxiv.org:1208.6091v1 in 2012).

Partitioning triples and partially ordered sets

Proc. Amer. Math. Soc., vol. 136, no. 5, (2008), 1823-1830

This article extends previous results from ω1 _to non-special partial orderings (including non-special trees), plus a little bit more. In particular, it demonstrates that if P → (ω)¹_ω, then (a) P → (ω + m, n)³ for all m, n < ω and (b) P → (ω + ω + 1, 4)³. This incidentally provides a marginally simpler proof of the result (due to E. C. Milner and K. Prikry) that ω₁ → (ω + ω + 1, 4)³.

A polarized partition relation for cardinals of countable cofinality

Proc. Amer. Math. Soc., vol. 136, no. 4, (2008), 1445-1449

The main result is that if cf κ = ω and λ = 2^<κ, then (λ : λ⁺) → (λ : λ⁺, (κ : α)_m)^1,1 for all α < ω₁ and m < ω. This generalizes and improves an old result of P. Erdős, A. Hajnal, and R. Rado. This result is somewhat sharp, in that it is known that under GCH the relation (κ : κ⁺) → (κ : κ⁺, κ : ω)^1,1 fails for all infinite κ with cf κ > ω.

More on partitioning triples of countable ordinals

Proc. Amer. Math. Soc., vol. 135, no. 4, (2007), 1197-1204

This paper documents more progress on the ω₁ → (α, n)³ problem. The main result is that the ordinary partition relation ω₁ → (ω + m, n)³ holds for all m, n < ω.

A polarized partition relation for weakly compact cardinals using elementary substructures

J. Symb. Logic, vol. 71, no. 4 (2006), 1342-1352

The following result is typical of those found in this article. If κ is weakly compact, then (κ : κ⁺) → (κ : α)^1,1_m and (κ : κ⁺) → (κ : κ^m)^1,1_μ for all α < κ⁺, m < ω, and μ < κ.

A brief remark on van der Waerden spaces

Proc. Amer. Math. Soc., vol. 132, no. 8 (2004), 2457-2460

This article provides a proof that MA(σ-centered) implies the existence of a van der Waerden space that is not a Hindman space. The proof is adapted from one of M. Kojman and S. Shelah that such a space exists under CH.

Be warned that there is a minor typo (in the definition of IP-convergent) on the first page of this article as printed in PAMS that is corrected in this article as served here.

A polarized partition relation using elementary substructures

J. Symb. Logic, vol. 65, no. 4 (2000), 1491-1498

The new result is that if λ = 2^<κ then (λ^{+ :} λ⁺) → (λ^{+ :} α ∨ α : λ⁺, κ + 1 : κ + 1)^1,1 for all α < λ⁺. This is a generalization and improvement of an old result of P. Erdős, A. Hajnal, and R. Rado.

A short proof of a partition relation for triples

Elec. J. Comb., vol. 7, no. 1 (2000), R24

Principally, this article presents a significantly shorter proof of an old result of P. Erdős and R. Rado that if L is a linear order with L → (ω)1ω _but without ω₁ ≤ L, then L → (ω + m, 4)³ for all m < ω. It also proves a couple of related results. If P is a partial order and κ is an infinite cardinal, then (a) if P → (κ + 2, ω)³, then P → (ω)¹₂^κ and (b) if P → (κ + 1, 4)³, then P → (cf κ)¹_κ^.

Some results in the partition calculus

Doctoral Thesis, Dartmouth College, Hanover, NH, US (1999)

My thesis includes a variety of results in the partition calculus, some (but not all) of which have been published separately in papers listed above.