A combinatorial proof of the extension property for partial isometries

TL;DR

The paper provides an elementary, self-contained combinatorial proof that the class of all finite metric spaces has the extension property for partial isometries ($EPPA$). It develops a two-step construction: first building a finite edge-labelled graph $oldsymbol{B}_0$ that extends all partial automorphisms of a given finite space but may fail to be metric, then refining it by expansion and sparsification to remove obstructions and completing to a metric space $oldsymbol{B}$ via shortest-path completion. A key technical device is the elimination of induced non-metric cycles, organized inductively by cycle size and aided by a Möbius-strip-like unwinding; the final step uses shortest-path completion to guarantee metric structure and the required extension properties. The approach ties into Ramsey-expansion techniques and extends to broader classes, including those described by forbidden homomorphisms and with algebraic closures, while also clarifying coherent EPPA aspects and connecting to prior work (Hodkinson–Otto, Herwig–Lascar).

Abstract

We present a short and self-contained proof of the extension property for partial isometries of the class of all finite metric spaces.

Figures (1)

• Figure 1: Expansion of a non-metric cycle with longest edge $\ell$ to a "Möbius strip".

Theorems & Definitions (7)

• Theorem 1.1: Solecki solecki2005, Vershik vershik2008
• proof
• Proposition 3.1
• proof
• proof : Proof of Theorem \ref{['mainresult']}
• Remark
• Remark