图的一阶可定义性：量词秩的上界
The First Order Definability of Graphs: Upper Bounds for Quantifier Rank
摘要：
我们称一阶公式A区分一个图G和另一个图G'，如果A在G上为真而在G'上为假。设G和G'是非同构的，设D（G，G'）表示这类公式的最小量词秩.本文证明了：若G和G'具有相同的阶数n，则D（G，G'）\le（n+3）/2，它紧到加常数1.类似的问题被认为是有向图（更一般地说，为任意结构的最大关系arity 2）和k-一致超图。此外，我们研究定义公式，我们要求A区分G从任何其他非同构G '。
英文摘要：
We say that a first order formula A distinguishes a graph G from another graph G' if A is true on G and false on G'. Provided G and G' are non-isomorphic, let D(G,G') denote the minimal quantifier rank of a such formula. We prove that, if G and G' have the same order n, then D(G,G')\le(n+3)/2, which is tight up to an additive constant of 1. The analogous questions are considered for directed graphs (more generally, for arbitrary structures with maximum relation arity 2) and for k-uniform hypergraphs. Also, we study defining formulas, where we require that A distinguishes G from any other non-isomorphic G'.
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