非循环族的Helly数
Helly numbers of acyclic families
摘要：
空交集合族的Helly数是它的最大包含极小空交子族的大小。设F是任意局部弧连通拓扑空间Gamma的开子集的有限族。设对F的每个子群G，G的元素的交至多有r个连通分支，每个连通分支都是Q-同调胞腔。证明了F的Helly数至多为r（d_Gamma+1），其中d_Gamma是使得Gamma的每个开集在j维及更高维上具有平凡Q-同调的最小整数j. (In特别是d_{R^d} = d）。这个界限是最好的。我们证明，事实上，一个更强的定理，小的子家庭可能有超过r连接组件，每个可能与非平凡的低维同源。作为一个应用程序，我们得到了几个明确的界限Helly数的几何横截理论，只有特设几何证明以前已知的，在某些情况下，我们得到的界限是优于以前已知的。
英文摘要：
The Helly number of a family of sets with empty intersection is the size of its largest inclusion-wise minimal sub-family with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Gamma. Assume that for every sub-family G of F the intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(d_Gamma+1), where d_Gamma is the smallest integer j such that every open set of Gamma has trivial Q-homology in dimension j and higher. (In particular d_{R^d} = d). This bound is best possible. We prove, in fact, a stronger theorem where small sub-families may have more than r connected components, each possibly with nontrivial homology in low dimension. As an application, we obtain several explicit bounds on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known; in certain cases, the bound we obtain is better than what was previously known.
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