(C_{10}\) has positive Turán density in the hypercube

The \(n\)-dimensional hypercube \(Q_n\) is a graph with vertex set \(\{0,1\}^n\) and there is an edge between two vertices if they differ in exactly one coordinate. For any graph \(H\), define \(\text{ex}(Q_n,H)\) to be the maximum number of edges of a subgraph of \(Q_n\) without a copy of \(H\). In...

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Bibliographic Details
Published in:arXiv.org 2024-03
Main Authors: Grebennikov, Alexandr, João Pedro Marciano
Format: Article
Language:English
Subjects:
Apexes
Density
Graph theory
Hypercubes
Vertex sets
Online Access:Get full text
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Summary:The \(n\)-dimensional hypercube \(Q_n\) is a graph with vertex set \(\{0,1\}^n\) and there is an edge between two vertices if they differ in exactly one coordinate. For any graph \(H\), define \(\text{ex}(Q_n,H)\) to be the maximum number of edges of a subgraph of \(Q_n\) without a copy of \(H\). In this short note, we prove that for any \(n \in \mathbb{N}\) $$\text{ex}(Q_n, C_{10}) > 0.024 \cdot e(Q_n).$$ Our construction is strongly inspired by the recent breakthrough work of Ellis, Ivan, and Leader, who showed that "daisy" hypergraphs have positive Turán density with an extremely clever and simple linear-algebraic argument.
ISSN:2331-8422