发表机构
Institute of Algebra and Geometry, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院代数与几何研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究聚焦超立方体中图的Turán密度这一公开问题，针对分层图情形，证明对任意给定围长要求，均存在围长不小于该值且超立方体Turán密度为正的分层图，拓展了已知正Turán密度分层图的类型。
AI 中文摘要
对于图$H$，设$\operatorname{ex}(Q_n, H)$为$n$维超立方体$Q_n$的不含同构于$H$的子图的子图的最大边数。$H$在超立方体中的Turán密度记为$π_\square(H)$，其定义为$\lim_{n\rightarrow \infty} \operatorname{ex}(Q_n, H)/|E(Q_n)|$。对于一般的$H$，确定$π_\square(H)$仍是一个广泛未解决的问题。康伦（Conlon）发现了一大类在超立方体中Turán密度为零的图。在本注记中，我们研究$π_{\square} (H)>0$的情形。若图$H$不能嵌入超立方体的边层中，则$π_{\square} (H)\geq 1/2$，这一点可通过取$Q_n$的每隔一个边层看出。在分层图中，已知在超立方体中具有正Turán密度的图仅为包含长度为$6$或$10$的圈的图。我们证明，对任意$g \geq 3$，存在围长至少为$g$的分层图，其在超立方体中的Turán密度至少为$1/2$。
英文摘要
For a graph $H$, let $\operatorname{ex}(Q_n, H)$ be the largest number of edges in a subgraph of the hypercube $Q_n$ of dimension $n$ that contains no subgraph isomorphic to $H$. The Turán density of $H$ in a hypercube, denoted $π_\square(H)$, is defined as $\lim_{n\rightarrow \infty} \operatorname{ex}(Q_n, H)/|E(Q_n)|$. Determining $π_\square(H)$ remains a widely open question for general $H$. Conlon found a large class of graphs with zero Turán density in a hypercube. In this note, we address the case when $π_{\square} (H)>0$. If a graph $H$ is not embeddable in an edge-layer of a hypercube, then $π_{\square} (H)\geq 1/2$, as can be seen by taking every other edge layer of $Q_n$. Among the layered graphs, the only ones known to have positive Turán density in a hypercube are graphs containing cycles of length $6$ or $10$. We show that, for every $g \geq 3$, there is a layered graph of girth at least $g$ whose Turán density in a hypercube is at least $1/2$.
