For $b=\omega+2$, see

András Hajnal. *Some results and problems on set theory*, Acta Math. Acad. Sci. Hungar., **11**, (1960), 277–298. MR0150044 (27 #47).

In this paper, András shows that $\omega_1\to(\omega\cdot n,\omega\cdot 2)^2_2$ for all $n<\omega$. This is best possible, in the sense that if $\alpha$ is countable, then $\alpha\not\to(\omega,\omega+1)^2_2$.

He also shows that $\mathsf{CH}$ implies that $\omega_1\not\to(\omega_1,\omega+2)^2_2$. I seem to remember Stevo saying that the same negative relation follows from adding a Cohen real, though I do not have details of the proof.

On the other hand, Stevo showed that the following statement $(1)$ is consistent:
$$\omega_1\to(\omega_1,\alpha)^2_2\mbox{ for all countable ordinals }\alpha $$
(in fact, he established a significant strengthening). His argument uses proper forcing, and it follows that $\mathsf{PFA}$ implies (1) (though no large cardinals are needed to establish its consistency together with $\mathsf{MA}+2^{\aleph_0}=\aleph_2$). See

Stevo Todorcevic. *Forcing positive partition relations*, Trans. Amer. Math. Soc., **280 (2)**, (1983), 703–720. MR0716846 (85d:03102).

As Péter Komjáth pointed out in a comment, there is a generalization of András result I should mention: The Baumgartner-Hajnal theorem states that if $\alpha<\omega_1$, then
$$ \omega_1\to(\alpha)^2_2, $$
provably in $\mathsf{ZFC}$. This solves your problem, as long as $b$ is countable.

In fact, Baumgartner-Hajnal admits a significant generalization that was pursued by several authors, most notably Fred Galvin and Stevo: Say that a poset $\phi$ is *non-special* iff $\phi\to(\omega)^1_\omega$. Any such poset satisfies $\phi\to(\alpha)^2_2$ for any countable $\alpha$, and this is best possible, since $\phi\to(\omega,\omega+1)^2_2$ already implies that $\phi$ is non-special. See

James Baumgartner, and András Hajnal. *A proof (involving Martin's axiom) of a partition relation*, Fund. Math., **78 (3)**, (1973), 193–203. MR0319768 (47 #8310),

and

Stevo Todorcevic. *Partition relations for partially ordered sets*, Acta Math., **155 (1-2)**,(1985), 1–25. MR0793235 (87d:03126).

For the general case, $b$ an arbitrary ordinal, not much seems known. There is of course the Erdős-Rado theorem, that gives in particular that if $\kappa$ is regular, then
$$ (2^{<\kappa})^+\to(\kappa+1)^2_\mu \mbox{ for all }\mu<\kappa, $$
and this is best possible in that no ordinal smaller than $(2^{<\kappa})^+$ works here, even if $\mu=2$.

This has been somewhat extender by James Baumgartner, András, and Stevo. See

James E. Baumgartner, András Hajnal, and Stevo Todorcevic. *Extensions of the Erdős-Rado theorem*. In **Finite and infinite combinatorics in sets and logic (Banff, AB, 1991)**, pp. 1–17, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 411, Kluwer Acad. Publ., Dordrecht, 1993. MR1261193 (95c:03111).

There, they show that if $\kappa$ is regular and uncountable, then forall $k<\omega$ and $\xi<\log\kappa$ (where $\log\kappa$ is the least cardinal $\mu$ such that $2^\mu\ge\kappa$), we have
$$ (2^{<\kappa})^+\to(\kappa+\xi)^2_k. $$
They also show that, under the same assumption on $\kappa$, $(2^{<\kappa})^+\to(\rho,(\kappa+n)_k)^2$, where $\rho=\kappa^{\omega+2}+1$ (ordinal exponentiation) and, improving results of Saharon Shelah, (the only result here where the ordinal on the left is not a cardinal) for all $n<\omega$,
$$ (2^{<\kappa})^+·\omega\to(\kappa·n)^2_2. $$
(On the other hand, I do not see any evidence that the ordinal on the left is least possible, or that a cardinal does not work.)

A good reference for the state of the art on partition calculus is

András Hajnal, and Jean A. Larson. *Partition relations*. In **Handbook of set theory. Vols. 1, 2, 3**, Akihiro Kanamori, and Matthew Foreman, eds., pp. 129–213, Springer, Dordrecht, 2010.

What follows comes from this paper, to which I refer for additional references (the numbers listed in brackets are as in their bibliography). In page 152, they write:

It was already asked in the Erdős-Hajnal problem lists [12, 13] if the partition
relations $\omega_2\to(\alpha)^2_2$ were consistent for $\alpha<\omega_2$. Though there is nothing to refute such consistency, the results going in this direction are weak and rare.

They add that Richard Laver [$40$] showed, under appropriate assumptions of large cardinal character, that $\omega_2\to(\omega_1\cdot2+1,\alpha)^2_2$ for all $\alpha<\omega_2$. In general, Richard showed that $\kappa^+\to(\kappa\cdot2+1,\alpha)^2_2$ for all $\alpha<\kappa^+$, provided that $\kappa$ carries a *Laver ideal*, that is, a non-trivial, $\kappa$-complete ideal such that given $\kappa^+$ sets not in the ideal, there are $\kappa^+$ of them, the intersection of any fewer than $\kappa$ of these also not in the ideal.

For $\omega_2$, the best result I recall is due to Matthew Foreman and András Hajnal. In [$20$] they show, under appropriate large cardinal assumptions, that $\omega_2\to(\omega_1^2+1,\alpha)^2_2$ for all $\alpha<\omega_2$. They also proved that if $\kappa$ is measurable, there is an ordinal $\Omega(\kappa)$ smaller than $\kappa^+$, but closed under of ordinal addition, multiplication, exponentiation, and taking ﬁxed points of these operations, such that
$$ \kappa^+\to(\alpha)^2_m\mbox{ for all }\alpha<\Omega(\kappa).$$
For the precise definition of $\Omega(\kappa)$, see Section 5 of the Hajnal-Larson paper.

Finally, Shelah [$59$] showed that if $\kappa$ is strongly compact, $\lambda>\kappa$ is a cardinal, and either $\lambda$ is regular or $\mathrm{cf}(\lambda)\ge\kappa$, then
$$ (2^{<\lambda})^+\to(\lambda+\zeta)^2_\theta\mbox{ for all }\zeta,\theta<\kappa. $$

isdifferent from coloring points, but I take this as evidence that if $b$ is not a cardinal, the statement that the minimal such $a$ must be a cardinal is probably false in general. $\endgroup$ – Noah Schweber Aug 6 '13 at 4:06