Creating Subgraphs in Semi-Random Hypergraph Games
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...
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Main Authors | , , |
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Format | Journal Article |
Language | English |
Published |
28.09.2024
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Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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DOI: | 10.48550/arxiv.2409.19335 |