  We study the pigeonhole principle for $\Sigma_2$-definable injections with
domain twice as large as the codomain, and the weak K\"onig lemma for
$\Delta^0_2$-definable trees in which every level has at least half of the
possible nodes. We show that the latter implies the existence of $2$-random
reals, and is conservative over the former. We also show that the former is
strictly weaker than the usual pigeonhole principle for $\Sigma_2$-definable
injections.
