EPPA numbers of graphs

Abstract

If G is a graph, A and B its induced subgraphs, and f : A→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 eppa(G), is the smallest number of vertices of an EPPA-witness for G, and we put eppa(n)=max⁡{eppa(G) : |G| = n}. In this note we review the state of the area, prove several lower bounds (in particular, we show that eppa(n) ≥ 2n/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 Kk-free graphs.