Grünbaum, Barnette, and Reay in 1974 completed the characterization of the pairs $$(f_i,f_j)$$ ( f i , f j ) of face numbers of 4-dimensional polytopes. Here we obtain a complete characterization of the pairs of flag numbers $$(f_0,f_{03})$$ ( f 0 , f 03 ) for 4-polytopes. Furthermore, we describe the pairs of face numbers $$(f_0,f_{d-1})$$ ( f 0 , f d - 1 ) for d -polytopes; this description is complete for even $$d\ge 6$$ d ≥ 6 except for finitely many exceptional pairs that are “small” in a well-defined sense, while for odd d we show that there are also “large” exceptional pairs. Our proofs rely on the insight that “small” pairs need to be defined and to be treated separately; in the 4-dimensional case, these may be characterized with the help of the characterizations of the 4-polytopes with at most eight vertices by Altshuler and Steinberg (1984).

中文翻译：

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表征多胞体的面部和标志向量对

Grünbaum、Barnette 和 Reay 在 1974 年完成了 4 维多胞体面数对 $$(f_i,f_j)$$ ( fi , fj ) 的表征。在这里，我们获得了 4 多胞体的标志数对 $$(f_0,f_{03})$$ ( f 0 , f 03 ) 的完整特征。此外，我们描述了 d-polytopes 的面数对 $$(f_0,f_{d-1})$$ ( f 0 , fd - 1 )；这个描述对于偶数 $$d\ge 6$$ d ≥ 6 是完整的，除了有限的许多在明确意义上“小”的异常对，而对于奇数 d，我们表明也有“大”异常对. 我们的证明依赖于需要定义“小”对并单独处理的洞察力；在 4 维情况下，