It is known that Ramsey property is a kind of generalizition of pigeon hole principle, and some kinds of Ramsey properties have lots of equivalent forms.
We often deal with the case $a\rightarrow (b)^r\_c $,when $a,b,c$ are cardinals; however, if we consider the ordertype of the homogeneous set, the question becomes much more complicated.


For instance, we can prove that any two-coloring of $\omega\_1$ must have a homogeneous subset of ordertype $\omega+1$, rather than only stating that can have a countable homogeneous subset. And also it's easy to show that for any countable ordinal $\alpha$, $\alpha\rightarrow (\omega+1)^2\_2$ fails.


I wonder: what is the least ordinal $c$ such that any 2-coloring of $c$ must have a homogeneous set of ordertype $\omega+2$? And can one prove that for any ordinal $b$, the least ordinal $a$ with $a\rightarrow b^2\_2$ must be a cardinal?


In this case, the properties of such $a$ is quite different from Erdős cardinal, any tricks used to show an Erdős cardinal must be inaccessible fail here.



A problem about Ramsey Property
For $b=\omega+2$, see 



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> András Hajnal. *Some results and problems on set theory*, Acta Math. Acad. Sci. Hungar., **11**, (1960), 277–298. [MR0150044 (27 #47)](http://www.ams.org/mathscinet-getitem?mr=150044). 
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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



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> Stevo Todorcevic. *Forcing positive partition relations*, Trans. Amer. Math. Soc., **280 (2)**, (1983), 703–720. [MR0716846 (85d:03102)](http://www.ams.org/mathscinet-getitem?mr=716846). 
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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



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> 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)](http://www.ams.org/mathscinet-getitem?mr=319768),
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and 



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> Stevo Todorcevic. *Partition relations for partially ordered sets*, Acta Math., **155 (1-2)**,(1985), 1–25. [MR0793235 (87d:03126)](http://www.ams.org/mathscinet-getitem?mr=793235). 
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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



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> 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)](http://www.ams.org/mathscinet-getitem?mr=1261193).
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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 



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> 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.
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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:



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> 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. 
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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
 $$ (2^{<\lambda})^+\to(\lambda+\zeta)^2\_\theta\mbox{ for all }\zeta,\theta<\kappa. $$


