Creating Subgraphs in Semi-Random Hypergraph Games

TL;DR

This paper investigates the process of constructing specific subgraphs within semi-random hypergraph games, providing bounds and thresholds for various hypergraph structures, including cliques, paths, and cycles.

Contribution

It extends the understanding of subgraph creation thresholds in semi-random hypergraph processes for general hypergraphs, especially for the case where 2 ≤ r < s.

Findings

Established upper and lower bounds for subgraph thresholds in semi-random hypergraph processes.

Derived exact thresholds for paths and cycles, showing bounds are sometimes tight.

Improved bounds for clique subgraph construction in the semi-random hypergraph setting.

Abstract

The semi-random hypergraph process is a natural generalisation of the semi-random graph process, which can be thought of as a one player game. For fixed $r < s$, starting with an empty hypergraph on $n$ vertices, in each round a set of $r$ vertices $U$ is presented to the player independently and uniformly at random. The player then selects a set of $s-r$ vertices $V$ and adds the hyperedge $U \cup V$ to the $s$-uniform hypergraph. For a fixed (monotone) increasing graph property, the player's objective is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the case where the player's objective is to construct a subgraph isomorphic to an arbitrary, fixed hypergraph $H$. In the case $r=1$ the threshold for the number of rounds required was already known in terms of the degeneracy of $H$. In the case $2 \le r < s$, we give upper and lower bounds on this threshold for general $H$, and find further improved upper bounds for cliques in particular. We identify cases where the upper and lower bounds match. We also demonstrate that the lower bounds are not always tight by finding exact thresholds for various paths and cycles.