超图的凸维数与超单纯形Van Kampen-Flores定理
The convex dimension of hypergraphs and the hypersimplicial Van Kampen-Flores Theorem
AI总结：
本研究完全确定了完全$k$-一致超图的凸维数，解决了相关公开问题，给出了一类超图最大超边数的界，并推广了van Kampen-Flores定理，刻画了保持超单纯形骨架的投影。
摘要：
$k$-一致超图的凸维数是指满足以下条件的最小维度$d$：存在其顶点到$\u211d^d$的单射映射，使得所有超边的$k$-重心集处于凸位置。我们完全确定了完全$k$-一致超图的凸维数，解决了Halman、Onn和Rothblum提出的一个公开问题，此前他们已解决了完全图的该问题。我们还针对估计具有凸维数$d$、$n$个顶点的$k$-一致超图的最大超边数这一极值问题，给出了下界和上界。为证明这些结果，我们将其重新表述为保持超单纯形顶点的仿射投影问题。更一般地，我们完整刻画了保持超单纯形$i$维骨架的投影。特别地，我们得到了线性van Kampen-Flores定理的超单纯形推广：对每个$n$、$k$和$i$，我们确定了$(n,k)$-超单纯形可线性投影到哪些维度且保持其$i$维骨架。我们的结果可通过$k$-集和$(i,j)$-划分直接解释，且与在$k$个点集的Minkowski和中寻找大的凸独立子集问题密切相关。
英文摘要：
The convex dimension of a $k$-uniform hypergraph is the smallest dimension $d$ for which there is an injective mapping of its vertices into $\mathbb{R}^d$ such that the set of $k$-barycenters of all hyperedges is in convex position. We completely determine the convex dimension of complete $k$-uniform hypergraphs, which settles an open question by Halman, Onn and Rothblum, who solved the problem for complete graphs. We also provide lower and upper bounds for the extremal problem of estimating the maximal number of hyperedges of $k$-uniform hypergraphs on $n$ vertices with convex dimension $d$. To prove these results, we restate them in terms of affine projections that preserve the vertices of the hypersimplex. More generally, we provide a full characterization of the projections that preserve its $i$-dimensional skeleton. In particular, we obtain a hypersimplicial generalization of the linear van Kampen-Flores theorem: for each $n$, $k$ and $i$ we determine onto which dimensions can the $(n,k)$-hypersimplex be linearly projected while preserving its $i$-skeleton. Our results have direct interpretations in terms of $k$-sets and $(i,j)$-partitions, and are closely related to the problem of finding large convexly independent subsets in Minkowski sums of $k$ point sets.
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