A Note on Polychromatic Colorings of Shift-Chains

TL;DR

This paper explores the colorability of shift-chain hypergraphs, showing that large uniform shift-chains are properly 2-colorable, but some are not 3-colorable, highlighting limitations in polychromatic coloring.

Contribution

It introduces new results on the colorability thresholds of shift-chain hypergraphs, emphasizing the boundary between 2-colorability and non-3-colorability.

Findings

Large uniform shift-chains are properly 2-colorable.

Some shift-chains are not polychromatic 3-colorable.

The paper clarifies the limits of polychromatic colorings in shift-chains.

Abstract

We popularize the question whether, for $m$ large enough, all $m$-uniform shift-chain hypergraphs are properly $2$-colorable. On the other hand, we show that for every $m$ some $m$-uniform shift-chains are not polychromatic $3$-colorable.