  We consider two intimately related statistical mechanical problems on
$\mathbb{Z}^3$: (i) the tricritical behaviour of a model of classical unbounded
$n$-component continuous spins with a triple-well single-spin potential (the
$|\varphi|^6$ model), and (ii) a random walk model of linear polymers with a
three-body repulsion and two-body attraction at the tricritical theta point
(critical point for the collapse transition) where repulsion and attraction
effectively cancel. The polymer model is exactly equivalent to a supersymmetric
spin model which corresponds to the $n=0$ version of the $|\varphi|^6$ model.
For the spin and polymer models, we identify the tricritical point, and prove
that the tricritical two-point function has Gaussian long-distance decay,
namely $|x|^{-1}$. The proof is based on an extension of a rigorous
renormalisation group method that has been applied previously to analyse the
$|\varphi|^4$ and weakly self-avoiding walk models on $\mathbb{Z}^4$.
