Dependence of functions on their variables
Ilijas Farah
TL;DR
The paper analyzes when a function $f$ on a combinatorial cube $f: \prod_{i<d} X_i \to Y$ essentially depends on a single variable. It establishes a dichotomy: either $f$ is determined, up to partitions and coordinate projections, by at most one coordinate, or there exist infinite sequences $(x_m)$ and $(y_m)$ with $d=u\sqcup v$ that exhibit a nontrivial cross-coordinate dependence, as derived via Ramsey theory and ultrafilter/compactness arguments. The proof proceeds in the equal-sets case, uses a Ramsey construction on triples to produce homogeneous sets, and relies on the Čech–Stone compactification $\beta X$ and AC to carry the inductive argument from $d=2$ to higher arities. The work also discusses the necessity of Choice, model-theoretic aspects in ZF, and connections to Čech–Stone remainder phenomena and dimension-type results in topological combinatorics.
Abstract
In this note I present a readable version of the proof of my 2001 result, giving a sufficient and necessary condition for a function on a combinatorial cube to essentially (locally) depend on at most one variable (see the end of the paper for the motivation), as well as some limiting results.