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 Baumgartner-Hajnal-Todorčević theorem (that if P → (ω)1ω, then P → (α)2n 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.

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)1κ 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, then + : 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.


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 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 for all α < ω1 and n < ω.


On a result of Szemerédi [pdf]
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 This improves somewhat on the (apparently unpublished) result of Szemerédi that Martin's axiom implies that (ω : c) → (ω : c, ω : κ)1,1 for any cardinal κ 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 in 2011), and Open and solved problems in infinite combinatorics (published as in 2012).

Partitioning triples and partially ordered sets [pdf | pdf]
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)1ω, 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 [pdf | pdf]
Proc. Amer. Math. Soc., vol. 136, no. 4, (2008), 1445-1449
The main result is that if cf κ = ω and λ = 2, 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 [pdf | pdf]
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 [pdf]
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)1,1m and (κ : κ+) → (κ : κm)1,1μ 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), 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 [pdf | pdf]
J. Symb. Logic, vol. 65, no. 4 (2000), 1491-1498
The new result is that if λ = 2, then + : 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)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)12κ, and (b) if P → (κ + 1, 4)3, then P → (cf κ)1κ.

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.