Where Pigeonhole Principles meet K\"onig Lemmas

We study the pigeonhole principle for ${\mathrm{\Sigma }}_2$-definable injections with domain twice as large as the codomain, and the weak K"onig lemma for ${\mathrm{\Delta }}_2^0$-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 ${\mathrm{\Sigma }}_2$-definable injections.