The proof in the paper. Color a pair of index pairs $((i, i'), (j, j'))$ by the truth values of φ(āᵢ', c_{i,j}) and $ψ(b̄ⱼ', c_{i,j})$, and take homogeneous sets $I$, $J$ by the bipartite Ramsey lemma. Pair up consecutive elements of $I$ and $J$: with ūᵢ = āᵣ āᵣ' for the $2i$-th and $(2i+1)$-st elements $r < r'$ of $I$, likewise $w̄ⱼ$, and $d_{i,j} = c_{r,s}$, the exclusive or $φ(x̄₁, z) ⊕ φ(x̄₂, z)$ holds of ūᵢ', d_{i,j} exactly when $i' = i$: earlier pairs see two equal truth values by homogeneity, the pair itself sees true then false, and later pairs see false twice. The same holds for the columns. Then the formula $γ = ∃z (U(z) ∧ (φ ⊕ φ) ∧ (ψ ⊕ ψ))$, with $U$ coloring the grid points $d_{i, h(J)}$ for $i ∈ J$, encodes membership $i ∈ J$ for an injection $h$ of the subsets of $[k]$ into the columns. By Baldwin–Shelah this bounds $k$, hence the order of the pattern.

The argument is carried out for the graph classes of the archive and their monadic dependence, and transferred to the graph classes used here.