  In this work we shall use a bottom-up approach for obtaining viable
inflationary Einstein-Gauss-Bonnet models which are also compatible with the
GW170817 event. Specifically, we shall use a recently developed theoretical
framework in which we shall specify only the tensor-to-scalar ratio, in terms
of the $e$-foldings number. Starting from the tensor-to-scalar ratio, we shall
reconstruct from it the Einstein-Gauss-Bonnet theory which can yield such a
tensor-to-scalar ratio, finding the scalar potential and the Gauss-Bonnet
coupling scalar function as functions of the $e$-foldings number. Accordingly,
the calculation of the spectral index of the primordial scalar perturbations,
and of the tensor spectral index easily is greatly simplified and these
observational indices can easily be found. After presenting the general
formalism for the bottom-up reconstruction, we exemplify our findings by
presenting several Einstein-Gauss-Bonnet models of interest which yield a
viable inflationary phenomenology. These models have also an interesting common
characteristic, which is a blue tilted tensor spectral index. We also
investigate the predicted energy spectrum of the primordial gravitational waves
for these Einstein-Gauss-Bonnet models, and as we show, all the models yield a
detectable primordial wave energy power spectrum.
