有限射影平面的凸集不可表示性
Non-representability of finite projective planes by convex sets
摘要：
我们证明了不存在d使得所有有限射影平面都可以由R^d中的凸集表示，回答了Alon，Kalai，Matousek和Meshulam的一个问题。这里，如果P是具有直线l_1，.的射影平面，l_n，P在R^d中的凸集表示是凸集C_1，.，C_n，使得C_{i_1}，.，C_{i_k}具有公共点，当且仅当对应的线l_{i_1}，...，l_{i_k}在P中有一个共同点。证明结合了Pach的正分式选择引理和Alon关于有限射影平面“展开”的一个结果。作为推论，我们证明了对于每个$d$都有2-可折叠的单纯复形，它们不是d-可表示的，加强了Matousek和作者的结果。
英文摘要：
We prove that there is no d such that all finite projective planes can be represented by convex sets in R^d, answering a question of Alon, Kalai, Matousek, and Meshulam. Here, if P is a projective plane with lines l_1,...,l_n, a representation of P by convex sets in R^d is a collection of convex sets C_1,...,C_n in R^d such that C_{i_1},...,C_{i_k} have a common point if and only if the corresponding lines l_{i_1},...,l_{i_k} have a common point in P. The proof combines a positive-fraction selection lemma of Pach with a result of Alon on "expansion" of finite projective planes. As a corollary, we show that for every $d$ there are 2-collapsible simplicial complexes that are not d-representable, strengthening a result of Matousek and the author.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
