In Progress

A proof (involving elementary substructures) of a partition relation [pdf] (with Jean A. Larson) [DRAFT]
 This article provides a proof of the full
BaumgartnerHajnalTodorčević theorem (that if P
→ (ω)^{1}_{ω}, then P →
(α)^{2}_{n} for all α <
ω_{1} 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.
 To be clear, this delay is entirely my fault. I only hope Jean will forgive me, both for causing the delay and for posting this draft here in the mean time.
 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 ErdősRado 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.
Submitted
 Even more on partitioning triples of countable ordinals [pdf]
(accepted for publication in Proc. Amer. Math. Soc. on 30 Sep 2016)
 The new result is that
ω_{1} → (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 ω_{1}
→ (α, n)^{3} for all α < ω_{1} and n < ω.
Published
 On a result of Szemerédi [pdf]
J. Symb. Logic,
vol. 73, no. 3 (2008), 953956

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unpublished) result of Szemerédi that Martin's axiom implies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 [pdf  pdf]
Proc. Amer. Math. Soc.,
vol. 136, no. 5, (2008), 18231830

This article extends previous results from ω_{1} to
nonspecial partial orderings (including nonspecial 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
ω_{1} → (ω + ω + 1, 4)^{3}.
 A polarized partition relation for cardinals of countable
cofinality [pdf  pdf]
Proc. Amer. Math. Soc.,
vol. 136, no. 4, (2008), 14451449

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 <
ω_{1} 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 [pdf  pdf]
Proc. Amer. Math. Soc.,
vol. 135, no. 4, (2007), 11971204

This paper documents more progress on the ω_{1} →
(α, n)^{3} problem. The main result is that the
ordinary partition relation ω_{1} → (ω + m,
n)^{3} holds for all m, n < ω.
 A polarized partition relation for weakly compact cardinals using
elementary substructures [pdf]
J. Symb. Logic,
vol. 71, no. 4 (2006), 13421352

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 [pdf  pdf]
Proc. Amer. Math. Soc.,
vol. 132, no. 8 (2004), 24572460

This article provides a proof that MA(σcentered) 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 IPconvergent) 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
[pdf  pdf]
J. Symb. Logic,
vol. 65, no. 4 (2000), 14911498

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 [pdf  pdf]
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)^{1}_{ω} but without
ω_{1} ≤ 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)^{1}_{2κ}, and (b) if P
→ (κ + 1, 4)^{3}, then P → (cf
κ)^{1}_{κ}.
 Some results in the partition calculus [pdf]
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.

