  We study four families of consequences of Ramsey's Theorem from the viewpoint
of reverse mathematics. The first, which we call the Achromatic Ramsey Theorem,
is from a partition relation introduced by Erd\H{o}s, Hajnal and Rado: $\omega
\to [\omega]^r_{c,\leq d}$, which asserts that for every $f: [\omega]^r \to c$
there exists an infinite $H$ with $|f([H]^r)| \leq d$. The second and third are
the Free Set Theorem and the Thin Set Theorem, which were introduced by Harvey
Friedman. And the last is the Rainbow Ramsey Theorem. We show that, most
theorems from these families are quite weak, i.e., they are strictly weaker
than $\operatorname{ACA}_0$ over $\operatorname{RCA}_0$. Interestingly, these
families turn out to be closely related. We establish the so-called strong cone
avoidance property of most instances of the Achromatic Ramsey Theorem by an
induction of exponents, then apply this and a similar induction to obtain the
strong cone avoidance property of the Free Set Theorem. From the strong cone
avoidance property of the Achromatic Ramsey Theorem and the Free Set Theorem,
we derive the strong cone property of the Thin Set Theorem and the Rainbow
Ramsey Theorem. It follws easily that a theorem with the strong cone avoidance
property does not imply $\operatorname{ACA}_0$ over $\operatorname{RCA}_0$.
