Unions of intervals in codes based on powers of sets

TL;DR

This paper constructs dense collections of subsets in multi-dimensional grids with specific symmetric difference restrictions, addressing open questions related to interval unions and combinatorial structures.

Contribution

It demonstrates the existence of dense subset collections with symmetric difference constraints, providing new insights into longstanding combinatorial conjectures.

Findings

Existence of dense collections with symmetric difference limitations

Limits on tightenings of Alon's 2023 question

Implications for Gowers' 2009 conjecture

Abstract

We prove that for every integer $d \ge 2$ there exists a dense collection of subsets of $[n]^d$ such that no two of them have a symmetric difference that may be written as the $d$th power of a union of at most $\lfloor d/2 \rfloor$ intervals. This provides a limitation on reasonable tightenings of a question of Alon from 2023 and of a conjecture of Gowers from 2009, and investigates a direction analogous to that of recent works of Conlon, Kam\v{c}ev, Leader, R\"aty and Spiegel on intervals in the Hales-Jewett theorem.