  In this work we present the production of charged particles associated with
high-$p_{\rm T}$ trigger particles ($8<p_{\rm T}^{\rm trig.}<15$ GeV /$c$) at
mid-pseudorapidity in proton-proton collisions at $\sqrt{s}=5.02$\,TeV
simulated with the \textsc{PYTHIA 8} Monte Carlo model. The study is performed
as a function of the relative transverse activity classifier, $R_{\rm T}$,
which is the relative charged-particle multiplicity in the transverse region
($\pi/3 <|\Delta\phi| <2\pi/3$) of the di-hadron correlations, and it is
sensitive to the Multi-Parton Interactions. The evolution of the yield of
associated particles on both the toward and the away regions with $3\leq p_{\rm
T}^{\rm assoc.}<8$ GeV/$c$ as a function of $R_{\rm T}$ is investigated. We
propose a strategy which allows for the modelling and subtraction of the
Underlying Event (UE) contribution from the toward and the away regions in
challenging environments like those characterised by large $R_{\rm T}$. We
found that the signal in the away region becomes broader with increasing
$R_{\rm T}$. Contrarily, the yield increases with $R_{\rm T}$ in the toward
region. This effect is reminiscent of that seen in heavy-ion collisions, where
an enhancement of the yield in the toward region for $0-5\%$ central Pb--Pb
collisions at $\sqrt{s}_{\rm NN}=2.76$\,TeV was reported. To further understand
the role of the UE and additional jet activity, the transverse region is
divided into two one-sided sectors, "trans-max" and "trans-min" selected in
each event according to which region has larger or smaller charged particle
multiplicity. Based on this selection criterion, the observables are studied as
a function of $R_{\rm T}^{\rm max}$ and $R_{\rm T}^{\rm min}$, respectively.
The presented results have been published in J. Phys. G 48, no.1, 015007 (2020)
and Phys. Rev. D 104, no.1, 016017 (2021).
