  In this work we shall prove that the tensor spectral index of the primordial
tensor perturbations for GW170817-compatible Einstein-Gauss-Bonnet theories,
takes the approximate simplified form $n_{\mathcal{T}}\simeq
2\left(-1+\frac{1}{\lambda(\phi)} \right)\epsilon_1$ at leading order, with
$\lambda (\phi)$ being a function of the scalar field which depends on the
scalar field potential and the second derivative of the scalar-Gauss-Bonnet
coupling $\xi''(\phi)$. With our analysis we aim to provide a definitive
criterion for selecting Einstein-Gauss-Bonnet models that can provide a
blue-tilted inflationary phenomenology, by simply looking at the scalar
potential and the scalar-Gauss-Bonnet coupling. We shall prove this using two
distinct approaches and as we show the tilt of the tensor spectral index is
determined by the values of the potential $V(\phi)$ and of scalar-Gauss-Bonnet
coupling at first horizon crossing. Specifically the blue-tilted tensor
spectral index can occur when $\xi''(\phi_*)V(\phi_*)>0$ at first horizon
crossing.
