  We consider two combinatorial principles, ${\sf{ERT}}$ and ${\sf{ECT}}$. Both
are easily proved in ${\sf{RCA}}_0$ plus ${\Sigma^0_2}$ induction. We give two
proofs of ${\sf{ERT}}$ in ${\sf{RCA}}_0$, using different methods to eliminate
the use of ${\Sigma^0_2}$ induction. Working in the weakened base system
${\sf{RCA}}_0^*$, we prove that ${\sf{ERT}}$ is equivalent to ${\Sigma^0_1}$
induction and ${\sf{ECT}}$ is equivalent to ${\Sigma^0_2}$ induction. We
conclude with a Weihrauch analysis of the principles, showing ${\sf{ERT}}
{\equiv_{\rm W}} {\sf{LPO}}^* {<_{\rm W}}{{\sf{TC}}_{\mathbb N}}^* {\equiv_{\rm
W}} {\sf{ECT}}$.
