Vertex correction to pairing at a Van Hove singularity

We calculate in a k-averaging approximation, the vertex corrections for pairing at a Van Hove singularity mediated by an Einstein phonon. The vertex correction leads to a modest reduction in ${\mathit{T}}_{\mathit{c}}$, which can still reach the experimental values for realistic phonon couplings, and to a significant deepening of the dip in the isotope shift at the ${\mathit{T}}_{\mathit{c}}$ maximum. Interference between first- and second-order vertex graphs diminishes the effect of the Coulomb interaction on ${\mathit{T}}_{\mathit{c}}$. The isotope-shift dip is shallower for repulsive Coulomb interaction, and deeper in the case of an attractive interaction introduced to model an excitonic contribution to pairing.