AI 中文总结
研究不含\(k\)维恩图作为迹的集系最大规模，证明\(\text{ex}_{tr}(n,VD_k)=O_k(n^{2^k - 2k + 1})\)改进了界，对\(k\)归纳证明，还给出相关下界构造、显式界及固定均匀性迹结果。
AI 中文摘要
我们研究了不包含\(k\)维恩图（记为\(VD_k\)）作为迹的集系的最大规模。对于每个固定的\(k\geq3\)，我们证明\(\text{ex}_{tr}(n,VD_k)=O_k(n^{2^k - 2k + 1})\)，改进了直接的绍尔 - 谢拉界\(O_k(n^{2^k - 1})\)。特别地，对于\(k = 4\)，指数从\(15\)降至\(9\)。证明从基瓦什、利德、朗和瓦格纳关于\(VD_3\)的定理出发，对\(k\)进行归纳，其中每条添加的边会免费强制产生两个新的维恩区域。我们还记录了固定均匀性下维恩图的下界构造、\(4\)均匀\(3\)维恩问题的显式界以及松散三角形的固定均匀性迹结果。
英文摘要
We study the maximum size of a set system that contains no $k$-Venn diagram, denoted by $VD_k$, as a trace. For every fixed $k\ge 3$, we prove $\text{ex}_{tr}(n,VD_k)=O_k(n^{2^k-2k+1})$, improving the direct Sauer-Shelah bound $O_k(n^{2^k-1})$. In particular, for $k=4$ the exponent decreases from $15$ to $9$. The proof starts from the theorem of Keevash, Leader, Long and Wagner for $VD_3$ and uses induction on $k$ in which two new Venn regions are forced for free at each added edge. We also record lower-bound constructions for Venn diagrams in fixed uniformity, explicit bounds for the $4$-uniform $3$-Venn problem, and a fixed uniformity trace result for the loose triangle.
