Is it consistent with $\mathsf{ZF}$ that there is a cardinal $\kappa$ such that for every $F\colon[\kappa]^\omega\to2$ there is an infinite $X\subseteq\kappa$ such that $F\upharpoonright[X]^\omega$ is constant? (an homogeneous set for $F$)

Rowbottom's theorem says that infinite homogeneous subsets exist for every $F\colon[\kappa]^{<\omega}\to2$ for measurable $\kappa$ and it is a standard result that in $\mathsf{ZFC}$ for every infinite cardinal $\kappa$ there is $F\colon[\kappa]^\omega\to2$ that admits no infinite homogeneous set in $\kappa$

Wikipedia: Rowbottom cardinal

I see that some results about Rowbottom cardinals in ZF are mentioned in Howard-Rubin page 242.

But this seems to be different from you r questions.

It is different, but interesting nonetheless, I wasn't aware of the terminology "Rowbottom's cardinal"

My question is equivalent to asking whether $\kappa\to(\omega)^\omega_2$ is consistent with $\mathsf{ZF}$ for some infinite $\kappa$ using the partition relation

While $\mathsf{ZFC}$ proves $\kappa\not\rightarrow(\omega)^\omega_2$ for every infinite $\kappa$

I just posted this as a question on main if you're interested to see the answer (hopefully I'll get one!)