EPPA numbers of graphs

TL;DR

The paper surveys the landscape of EPPA numbers for graphs, defining EPPA-witnesses and the EPPA-number, and recalls Hrushovski’s foundational result that graphs have the extension property for partial automorphisms. It establishes a new, strong lower bound $eppa(n) \ge {n-1 \choose \lfloor (n-1)/2 \rfloor} = \Omega(2^n/\sqrt{n})$, intensifying the gap with the best known upper bounds and helping identify the exponential base. It then reviews three principal upper-bound constructions—finite homogeneous graphs, Kneser graphs, and valuation graphs—and extends the discussion to related structures such as hypergraphs and directed graphs, including recent refinements and coherent variants. The note also compiles extensive open questions and directions, ranging from questions about which classes have EPPA-witnesses (e.g., tournaments, Steiner systems) to finer bounds for specific graph families (cycles, planar graphs) and the interplay between standard and coherent EPPA numbers. Overall, the work highlights the exponential growth of EPPA witnesses in typical graphs while outlining promising lines of inquiry to close the gap between lower and upper bounds and to extend EPPA concepts to broader combinatorial structures.

Abstract

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 \emph{partial automorphism} of $G$. In 1992, Hrushovski proved that graphs have the \emph{extension property for partial automorphisms} (\emph{EPPA}, also called the \emph{Hrushovski property}), that is, for every finite graph $G$ there is a finite graph $H$, an \emph{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 $\mathop{\mathrm{eppa}}\nolimits(G)$, 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.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 2.2
• Theorem 2.3: Herwig--Lascar, 2000 herwig2000
• proof : Sketch of a proof
• Theorem 2.4: Evans--Hubička--Konečný--Nešetřil, 2018 eppatwographs
• proof : Sketch of a proof from Hubicka2018EPPA