If $G$ is a graph, $A,B$ its induced subgraphs and $f\colon A\to B$ an isomorphism, we say that $f$ is a partial automorphism of $G$. In 1992, Hrushovski proved that graphs have the extension property for partial automorphisms (EPPA, also called the Hrushovski property), that is, for every finite graph $G$ there is a finite graph $H$, its EPPA-witness, such that $G$ is an induced subgraph of $H$ and every partial automorphism of $G$ extends to an automorphism of $H$. The EPPA number of a graph $G$, denoted by $\mathop{\mathrm{eppa}}\nolimits(G)$, is the smallest number of vertices of an EPPA-witness for $G$, and we put $\mathop{\mathrm{eppa}}\nolimits(n) = \max\{\mathop{\mathrm{eppa}}\nolimits(G) : \lvert G\rvert = n\}$. In this note we review the state of the area, improve some lower bounds (in particular, we show that $\mathop{\mathrm{eppa}}\nolimits(n)\geq \frac{2^n}{\sqrt{n}}$, thereby identifying the correct base of the exponential) and pose several open questions. We also briefly discuss EPPA numbers of hypergraphs and directed graphs.

翻译：设$G$为图，$A,B$为其诱导子图，且$f\colon A\to B$为同构映射，则称$f$是$G$的部分自同构。1992年，Hrushovski证明了图具有部分自同构的延拓性质（EPPA，亦称Hrushovski性质），即对任意有限图$G$，存在有限图$H$（称为$G$的EPPA-见证），使得$G$为$H$的诱导子图，且$G$的每个部分自同构均可延拓为$H$的自同构。图$G$的EPPA数记为$\mathop{\mathrm{eppa}}\nolimits(G)$，定义为$G$的EPPA-见证的最小顶点数，并记$\mathop{\mathrm{eppa}}\nolimits(n) = \max\{\mathop{\mathrm{eppa}}\nolimits(G) : \lvert G\rvert = n\}$。本文回顾该领域的研究现状，改进若干下界（特别地，我们证明$\mathop{\mathrm{eppa}}\nolimits(n)\geq \frac{2^n}{\sqrt{n}}$，从而确定了指数的正确底数），并提出若干未解决问题。此外，简要讨论超图与有向图的EPPA数。