EPPA numbers of graphs

Abstract

If G is a graph, A and B its induced subgraphs, and $f:A\to B$ 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, an EPPA-witness for G, 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 $\mathrm{eppa}\left(G\right)$, is the smallest number of vertices of an EPPA-witness for G, and we put $\mathrm{eppa}\left(n\right)=\max \left\{\mathrm{eppa}\right(G):|G|=n\}$. In this note we review the state of the area, prove several lower bounds (in particular, we show that $\mathrm{eppa}\left(n\right)\ge \frac{2^n}{\sqrt{n}}$, 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.