Arbitrary-width non-trivial F-sets over every finite field

Establish that, for every finite field \(\mathbb{F}_q\) and every positive integer \(r\), there exists a non-trivial \(F\)-set of width exactly \(r\).

Background

Ferraguti and Micheli conjectured the existence of non-trivial FF-sets of arbitrary width over every finite field. The paper proves the finite-width assertion for every q≠2,3q\neq2,3, thereby leaving the cases of the fields F2\mathbb{F}_2 and F3\mathbb{F}_3 unresolved.

References

They concluded by conjecturing that non-trivial $F$-sets of arbitrary width exist over every finite field Conjecture~5.2.

— F-sets of arbitrary finite width  (2609.05305 - Giannoni, 4 Sep 2026) in Section 1, Introduction

The argument does not prove non-existence of $F$-sets of positive finite width over $_2$ or $_3$.

— F-sets of arbitrary finite width  (2609.05305 - Giannoni, 4 Sep 2026) in Section 5, concluding discussion after the proof of Theorem 3.4