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Creating Subgraphs in Semi-Random Hypergraph Games

Natalie Behague, Pawel Pralat, Andrzej Rucinski

TL;DR

This work extends the semi-random process to $s$-uniform hypergraphs with a random $r$-set drawn each step and a player choosing the remaining $s-r$ vertices to form edges, and investigates the thresholds $\tau^{(r)}(H)$ for embedding a fixed $s$-graph $H$. It proves a general lower bound $\tau^{(r)}(H) \ge n^{r-(k-s+r)/m}$ (with $k=|V(H)|$, $m=|E(H)|$) and develops sharp upper bounds for structured targets such as $(s,s-r)$-starpluses and certain cliques, via explicit strategies and probabilistic tools including the second moment method and model transitions. The paper also provides improved lower bounds for $\ell$-tight paths and cycles and computes explicit thresholds in key cases like $K_6^{(3)}$ and other $K_k^{(3)}$ using a multi-phase construction with rooted subgraphs. The results advance understanding of how the degeneracy-like and star-like structures govern threshold behavior in semi-random hypergraph processes, and raise several open questions about tight thresholds for broader families of hypergraphs.

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.

Creating Subgraphs in Semi-Random Hypergraph Games

TL;DR

This work extends the semi-random process to -uniform hypergraphs with a random -set drawn each step and a player choosing the remaining vertices to form edges, and investigates the thresholds for embedding a fixed -graph . It proves a general lower bound (with , ) and develops sharp upper bounds for structured targets such as -starpluses and certain cliques, via explicit strategies and probabilistic tools including the second moment method and model transitions. The paper also provides improved lower bounds for -tight paths and cycles and computes explicit thresholds in key cases like and other using a multi-phase construction with rooted subgraphs. The results advance understanding of how the degeneracy-like and star-like structures govern threshold behavior in semi-random hypergraph processes, and raise several open questions about tight thresholds for broader families of hypergraphs.

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 , starting with an empty hypergraph on vertices, in each round a set of vertices is presented to the player independently and uniformly at random. The player then selects a set of vertices and adds the hyperedge to the -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 . In the case the threshold for the number of rounds required was already known in terms of the degeneracy of . In the case , we give upper and lower bounds on this threshold for general , 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.
Paper Structure (15 sections, 20 theorems, 144 equations, 15 figures)

This paper contains 15 sections, 20 theorems, 144 equations, 15 figures.

Table of Contents

  1. Introduction
  2. New results
  3. Lower bound
  4. Upper bounds which match lower bound
  5. Upper bounds for general $s$-graphs and cliques
  6. Better lower bounds
  7. Probabilistic tools
  8. Proof of Theorem \ref{['thm:lower_bound_general']}
  9. Proof of Theorem \ref{['thm:upper_bound_balanced']}
  10. Proof of Theorem \ref{['thm:general_cliques']}
  11. The clique $K_6^{(3)}$
  12. Larger 3-uniform cliques ($r=2$)
  13. General cliques
  14. Proof of Proposition \ref{['prop:rsl']}
  15. Open Questions

Key Result

Theorem 1

Let $s\ge2$ and $H$ be a fixed $s$-uniform hypergraph of degeneracy $d \in {\mathbb N}$. Then, $\tau^{(1)}(H) = n^{1-1/d}$.

Figures (15)

  • Figure 1: The $3$-graph $H$ described in Example \ref{['Ex1']}.
  • Figure 2: The $(4,2)$-starplus $H$ described in Example \ref{['starplus']}, where the black solid edges are the edges of the $2$-star $S$ and the red dashed edges are the excess edges.
  • Figure 3: The wheel $W_8^{(5,1)}$, with the 'ray' edges containing the centre in blue and the 'excess' edges of the cap in red.
  • Figure 4: An example of the graphs $H'$ and $\widehat{H}$ constructed in the proof of Corollary \ref{['cor:upper_bound']} for a given $H$. The red dashed edges represent the edges of $H'$.
  • Figure 5: A $3$-tight $5$-uniform cycle $C^{(5,3)}_6$ and a $3$-tight $5$-uniform path $P^{(5,3)}_4$.
  • ...and 10 more figures

Theorems & Definitions (41)

  • Theorem 1: Behague, Marbach, Prałat, Ruciński BMPR
  • Theorem 2
  • Example 1
  • Corollary 3
  • Example 2
  • Example 3
  • Theorem 4
  • Example 4
  • Corollary 5
  • Example 5
  • ...and 31 more