Large monochromatic components and long monochromatic cycles in random hypergraphs

TL;DR

This paper investigates the size of monochromatic components and long monochromatic cycles in randomly colored hypergraphs, extending classical results from complete hypergraphs to the probabilistic setting.

Contribution

It generalizes known extremal hypergraph results to random hypergraphs, providing new bounds on monochromatic structures in this probabilistic context.

Findings

Largest monochromatic component sizes in random hypergraphs are characterized.

Long monochromatic loose cycles are shown to exist with high probability.

Results extend classical deterministic hypergraph theorems to random settings.

Abstract

We extend results of Gy\'arf\'as and F\"uredi on the largest monochromatic component in $r$-colored complete $k$-uniform hypergraphs to the setting of random hypergraphs. We also study long monochromatic loose cycles in $r$-colored random hypergraphs. In particular, we obtain a random analog of a result of Gy\'arf\'as, S\'ark\"ozy, and Szemer\'edi on the longest monochromatic loose cycle in $2$-colored complete $k$-uniform hypergraphs.