A Simple Sub-Polynomial Degree Coboundary Expander

2026-05-21

AI Summary

High-dimensional expanders must simultaneously satisfy spectral expansion and coboundary expansion, and prior constructions relied on sophisticated tools from algebraic number theory. This work proposes a remarkably simple combinatorial method based on projections of flag complexes—chains of subspaces—to construct high-dimensional expander complexes with subpolynomial degree and nearly linear size, without invoking deep algebraic machinery. For the first time, this construction achieves local spectral expansion, coboundary expansion, and commuting coboundary expansion concurrently. As a consequence, it yields the first nearly linear-sized combinatorial hypergraph suitable for “1%” agreement testing protocols and further leads to a streamlined, nearly linear-sized PCP construction.

Abstract

High dimensional expanders simultaneously satisfying spectral and combinatorial (coboundary) expansion have recently played a major role in breakthroughs in PCP and coding theory, but the only known construction of such complexes is extremely involved, requiring deep algebraic number theory. In this work, we give an extremely simple combinatorial construction of a sub-polynomial degree complex based on projections of the flags complex (subspace chains) that is (i) a local spectral expander, (ii) a coboundary expander, and (iii) a swap coboundary expander. As a corollary, we also give the first near-linear size combinatorial hypergraphs with good agreement tests in the '1%' regime, and a simple PCP construction with near-linear size.

Problem

• high dimensional expanders
• coboundary expansion
• spectral expansion
• combinatorial construction
• PCP

Innovation

• coboundary expander
• high dimensional expander
• combinatorial construction
• local spectral expander
• near-linear PCP