  We use the framework of reverse mathematics to address the question of, given
a mathematical problem, whether or not it is easier to find an infinite partial
solution than it is to find a complete solution. Following Flood, we say that a
Ramsey-type variant of a problem is the problem with the same instances but
whose solutions are the infinite partial solutions to the original problem. We
study Ramsey-type variants of problems related to K\"onig's lemma, such as
restrictions of K\"onig's lemma, Boolean satisfiability problems, and graph
coloring problems. We find that sometimes the Ramsey-type variant of a problem
is strictly easier than the original problem (as Flood showed with weak
K\"onig's lemma) and that sometimes the Ramsey-type variant of a problem is
equivalent to the original problem. We show that the Ramsey-type variant of
weak K\"onig's lemma is robust in the sense of Montalban: it is equivalent to
several perturbations of itself. We also clarify the relationship between
Ramsey-type weak K\"onig's lemma and algorithmic randomness by showing that
Ramsey-type weak weak K\"onig's lemma is equivalent to the problem of finding
diagonally non-recursive functions and that these problems are strictly easier
than Ramsey-type weak K\"onig's lemma. This answers a question of Flood.
