Establish a finitary version of the dependence theorem

Determine whether, for every pair of positive integers d and k, there exists an integer N=N(d,k) such that every function f:X^d\to X with no finite configurations \(d=u\sqcup v\), \(x_m\in X^u\), and \(y_m\in X^v\) for \(m<N\) satisfying \(f(x_l^\frown y_l)\ne f(x_m^\frown y_n)\) for all \(l<m<n<N\) admits a partition of X into k pieces on each associated rectangle of which f depends on at most one coordinate.

Background

The main dependence theorem is infinitary: if a function fails to be locally dependent on at most one variable after every finite partition of each coordinate set, then it yields infinite sequences witnessing a strong two-block dependence pattern. The author asks whether this can be replaced by a finitary criterion with a bound depending only on the dimension d and the number k of partition pieces.

The proposed question specifies the exact finite obstruction and the desired conclusion. A positive answer would provide a uniform finite witness for the dichotomy between strong dependence and local one-variable dependence, but the paper does not resolve whether such a bound N(d,k) exists.

References

In Question~11 I asked whether there is a finitary version of Theorem~\ref{T.Dependence}, without suggesting what such finitary version should look like. A more precise question (to which I do not know the answer) is whether for all $d$ and $k$ there is $N=N(d,k)$ such that for all $X$ and all $f\colon Xd\to X$, if there are no $d=u\sqcup v$, $x_m\in Xu$, $y_m\in Yv$ for $m<n$ such that $f(x_l{}\frown y_l)\neq f(x_m{}\frown y_n)$ for all $ l<N$ and $m<n<N$, then there is a partition $X=\bigsqcup_{j<k} U_{i}$ such that for every $s\in kd$ there are $j(s)<d$ and $g_s\colon U_{j(s)}\to Y$ such that $f$ agrees with $g_s\circ \pi_{j(s)}$ on $\prod_{i<d} U_{s(i)}$.

— Dependence of functions on their variables  (2503.07864 - Farah, 10 Mar 2025) in Section 3, “Concluding remarks,” final paragraph before the motivation discussion