On a common-extendable, non-Sidorenko linear system
Abstract
A system of linear equations in \(\mathbb{F}_p^n\) is common if every two-colouring of \(\mathbb{F}_p^n\) yields at least as many monochromatic solutions as a random two-colouring, asymptotically as \(n \to \infty\). By analogy to the graph-theoretic setting, Alon has asked whether any (non-Sidorenko) system of linear equations can be made uncommon by adding sufficiently many free variables. Fox, Pham and Zhao answered this question in the affirmative among systems which consist of a single equation. We answer Alon's question in the negative.
We also observe that the property of remaining common despite that addition of arbitrarily many free variables is closely related to a notion of commonness in which one replaces the arithmetic mean of the number of monochromatic solutions with the geometric mean, and furthermore resolve questions of Kamčev-Liebenau-Morrison.
Mathematics Subject Classifications: 05D10, 11B30
Keywords: Sidorenko's conjecture, Sidorenko and common linear patterns