the Ramsey number of [itex] R(\omega,\omega)=\omega [/itex] but then [itex] R(\omega+1,\omega)=\omega_1 [/itex] My question is on the second one can we do a counter example to show that it cant be any countable ordinal. depending on how I count the natural numbers I can get any countable ordinal I want. If we assume that [itex] R(\omega+1,\omega) [/itex] was equal to some countable ordinal then I could just color [itex] \omega [/itex] edges with blue for example and i would stay under the [itex] \omega+1 [/itex] limit . And the other color we would just use a finite number of them. I guess i don't really understand why order matters for an infinite Ramsey graph. It doesn't seem like it matters in the finite case.