平面图的几乎线性通用点集
Almost Linear Universal Point Sets for Planar Graphs
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中文总结 AI 辅助
针对平面图通用点集，提出大小为 n^{1+o(1)} 的构造，利用排列超模式与有序根森林，改进二次上界至几乎线性。
中文摘要 AI 辅助
一个点集对 $n$ 个顶点的平面图是通用的，如果每个这样的图都存在一个无交叉的直线绘制，其顶点属于该点集。我们构造了大小为 $n^{1+o(1)}$ 的通用点集，改进了之前的二次上界。我们的构造使用了 Bannister、Cheng、Devanny 和 Eppstein 从通用点集到 $213$-避免排列的超模式的归约。我们将这些排列表示为有序根森林，并构造一个包含每个此类森林的小区间族。结果由区间族大小的一个直接界得出。GPT-6 Astra 协助开发了构造和证明。
英文摘要
A point set is universal for planar graphs on $n$ vertices if every such graph has a straight-line drawing without crossings whose vertices belong to the set. We construct universal point sets of size $n^{1+o(1)}$, improving the previous quadratic upper bound. Our construction uses the reduction of Bannister, Cheng, Devanny, and Eppstein from universal point sets to superpatterns for $213$-avoiding permutations. We represent these permutations by ordered rooted forests and construct a small family of intervals containing every such forest. The result follows from a straightforward bound on the size of the family of intervals. GPT-6 Astra assisted in developing the construction and proof.
