If $G$ is a graph, $A$ and $B$ 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, prove several 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 many open questions. We also briefly discuss EPPA numbers of hypergraphs, directed graphs, and $K_k$-free graphs.

翻译：设$G$是一个图，$A$和$B$是其诱导子图，且$f\colon A\to B$是一个同构，则称$f$为$G$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，也称为Hrushovski性质），即对每个有限图$G$，存在一个有限图$H$（称为$G$的EPPA-见证图），使得$G$是$H$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，记作$\mathop{\mathrm{eppa}}\nolimits(G)$，是$G$的EPPA-见证图中顶点数的最小值，并定义$\mathop{\mathrm{eppa}}\nolimits(n) = \max\{\mathop{\mathrm{eppa}}\nolimits(G) : \lvert G\rvert = n\}$。本文回顾该领域的研究现状，证明若干下界（特别地，我们得到$\mathop{\mathrm{eppa}}\nolimits(n)\geq \frac{2^n}{\sqrt{n}}$，从而确定了指数基的正确形式），并提出许多未解决问题。我们还将简要讨论超图、有向图和$K_k$-自由图的EPPA数。