Ramsey and Erdős–Rado: exact orientation obligations

The local results establish two different partition bounds in ZFC:

$\omega ⟶(\omega {)}_r^n(1\le n,r<\omega ),$

by Infinite Ramsey theorem for fixed finite arity and colors, and

${\mathrm{\beth }}_n(\kappa {)}^{+}⟶({\kappa }^{+}{)}_{\kappa }^{n+1}(\kappa \text{ infinite},n<\omega ),$

by Erdős–Rado for arbitrary infinite cardinals and finite arity. The meanings of homogeneous and the cardinal arrow are those of Partition arrows and homogeneous sets. The second theorem uses the relative finite beths beginning at $\kappa$, and includes the singleton-coloring argument at $n=0$.

Each statement fixes its finite arity before quantifying colorings. Neither says that one infinite set is simultaneously homogeneous for all finite arities. The first theorem has finitely many colors; the second permits $\kappa$ colors by enlarging the ambient cardinal to the indicated beth successor. In the latter proof the end-homogeneous sequence need not increase as ambient ordinals; the final reduction uses the largest index. Monk's source statement gives the countable-color case, while the local theorem supplies the stated arbitrary-cardinal argument. No later partition theorem is being used as a prerequisite.