  A graph $F$ is Ramsey for a pair of graphs $(G,H)$ if any red/blue-coloring
of the edges of $F$ yields a copy of $G$ with all edges colored red or a copy
of $H$ with all edges colored blue. Two pairs of graphs are called Ramsey
equivalent if they have the same collection of Ramsey graphs. The symmetric
setting, that is, the case $G=H$, received considerable attention. This led to
the open question whether there are connected graphs $G$ and $G'$ such that
$(G,G)$ and $(G',G')$ are Ramsey equivalent. We make progress on the asymmetric
version of this question and identify several non-trivial families of Ramsey
equivalent pairs of connected graphs.
  Certain pairs of stars provide a first, albeit trivial, example of Ramsey
equivalent pairs of connected graphs. Our first result characterizes all Ramsey
equivalent pairs of stars. The rest of the paper focuses on pairs of the form
$(T,K_t)$, where $T$ is a tree and $K_t$ is a complete graph. We show that, if
$T$ belongs to a certain family of trees, including all non-trivial stars, then
$(T,K_t)$ is Ramsey equivalent to a family of pairs of the form $(T,H)$, where
$H$ is obtained from $K_t$ by attaching disjoint smaller cliques to some of its
vertices. In addition, we establish that for $(T,H)$ to be Ramsey equivalent to
$(T,K_t)$, $H$ must have roughly this form. On the other hand, we prove that
for many other trees $T$, including all odd-diameter trees, $(T,K_t)$ is not
equivalent to any such pair, not even to the pair $(T, K_t\cdot K_2)$, where
$K_t\cdot K_2$ is a complete graph $K_t$ with a single edge attached.
