The<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>e</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>e</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo>→</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>K</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>K</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mn>2</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>cross sections at center-of-mass energies from production threshold to 4.5 GeV measured with initial-state radiation

We study the processes ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}3({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})\ensuremath{\gamma}$, $2({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0})\ensuremath{\gamma}$ and ${K}^{+}{K}^{\ensuremath{-}}2({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})\ensuremath{\gamma}$, with the photon radiated from the initial state. About 20 000, 33 000 and 4000 fully reconstructed events, respectively, have been selected from $232\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of BABAR data. The invariant mass of the hadronic final state defines the effective ${e}^{+}{e}^{\ensuremath{-}}$ center-of-mass energy, so that these data can be compared with the corresponding direct ${e}^{+}{e}^{\ensuremath{-}}$ measurements. From the $3({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})$, $2({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0})$ and ${K}^{+}{K}^{\ensuremath{-}}2({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})$ mass spectra, the cross sections for the processes ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}3({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})$, ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}2({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0})$ and ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}2({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})$ are measured for center-of-mass energies from production threshold to 4.5 GeV. The uncertainty in the cross section measurement is typically 6%--15%. We observe a structure at 1.9 GeV in both cross sections and a resonance structure with mass $1645\ifmmode\pm\else\textpm\fi{}0.008\text{ }\text{ }\mathrm{GeV}/{\mathrm{c}}^{2}$ and width $0.114\ifmmode\pm\else\textpm\fi{}0.014\text{ }\text{ }\mathrm{GeV}$ when the $\ensuremath{\omega}(782)\ensuremath{\eta}$ final state is extracted. We observe the $J/\ensuremath{\psi}$ in all these final states and measure the corresponding branching fractions.