  The production rates and the transverse momentum distribution of strange
hadrons at mid-rapidity ($\ |y\ | < 0.5$) are measured in proton-proton
collisions at $\sqrt{s}$ = 13 TeV as a function of the charged particle
multiplicity, using the ALICE detector at the LHC. The production rates of
$\rm{K}^{0}_{S}$, $\Lambda$, $\Xi$, and $\Omega$ increase with the multiplicity
faster than what is reported for inclusive charged particles. The increase is
found to be more pronounced for hadrons with a larger strangeness content.
Possible auto-correlations between the charged particles and the strange
hadrons are evaluated by measuring the event-activity with charged particle
multiplicity estimators covering different pseudorapidity regions. When
comparing to lower energy results, the yields of strange hadrons are found to
depend only on the mid-rapidity charged particle multiplicity. Several features
of the data are reproduced qualitatively by general purpose QCD Monte Carlo
models that take into account the effect of densely-packed QCD strings in high
multiplicity collisions. However, none of the tested models reproduce the data
quantitatively. This work corroborates and extends the ALICE findings on
strangeness production in proton-proton collisions at 7 TeV.
