  Often regarded as the study of how order emerges from randomness, Ramsey
theory has played an important role in mathematics and computer science, giving
rise to applications in numerous domains such as logic, parallel processing,
and number theory. The core of graph Ramsey theory is arrowing: For fixed
graphs $F$ and $H$, the $(F, H)$-Arrowing problem asks whether a given graph,
$G$, has a red/blue coloring of the edges of $G$ such that there are no red
copies of $F$ and no blue copies of $H$. For some cases, the problem has been
shown to be coNP-complete, or solvable in polynomial time. However, a more
systematic approach is needed to categorize the complexity of all cases.
  We focus on $(P_3, H)$-Arrowing as $F = P_3$ is the simplest meaningful case
for which the complexity question remains open, and the hardness for this case
likely extends to general $(F, H)$-Arrowing for nontrivial $F$. In this
pursuit, we also gain insight into the complexity of a class of matching
removal problems, since $(P_3, H)$-Arrowing is equivalent to $H$-free Matching
Removal. We show that $(P_3, H)$-Arrowing is coNP-complete for all
$2$-connected $H$ except when $H = K_3$, in which case the problem is in P. We
introduce a new graph invariant to help us carefully combine graphs when
constructing the gadgets for our reductions. Moreover, we show how
$(P_3,H)$-Arrowing hardness results can be extended to other $(F,H)$-Arrowing
problems. This allows for more intuitive and palatable hardness proofs instead
of ad-hoc constructions of SAT gadgets, bringing us closer to categorizing the
complexity of all $(F, H)$-Arrowing problems.
