How many random edges make a dense hypergraph non-2-colorable?

TL;DR

This paper establishes a precise threshold for the number of random edges needed to make a dense hypergraph non-2-colorable, advancing understanding of phase transitions in hypergraph colorability.

Contribution

The authors derive a tight bound on the number of random edges required to induce non-2-colorability in dense hypergraphs, connecting edge density with probabilistic colorability thresholds.

Findings

Adding omega(n^{k epsilon/2}) random edges makes hypergraphs non-2-colorable.

Hypergraphs with Omega(n^{k-epsilon}) edges become non-2-colorable after sufficient random edges.

The bounds are tight, with existing hypergraphs remaining 2-colorable below the threshold.

Abstract

We study a model of random uniform hypergraphs, where a random instance is obtained by adding random edges to a large hypergraph of a given density. We obtain a tight bound on the number of random edges required to ensure non-2-colorability. We prove that for any k-uniform hypergraph with Omega(n^{k-epsilon}) edges, adding omega(n^{k epsilon/2}) random edges makes the hypergraph almost surely non-2-colorable. This is essentially tight, since there is a 2-colorable hypergraph with Omega(n^{k-\epsilon}) edges which almost surely remains 2-colorable even after adding o(n^{k \epsilon / 2}) random edges.