Geodesic
On homogeneous extensions of finite predicate systems
Fundamentalʹnaâ i prikladnaâ matematika, Tome 15 (2009) no. 7, pp. 137-140.

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It is shown that any finite $n$-element predicate system for which any two one-element subsystems are isomorphic is embeddable in a finite $(2^n-1)$-element system having a transitive automorphism group. 
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E. V. Ovchinnikova. On homogeneous extensions of finite predicate systems. Fundamentalʹnaâ i prikladnaâ matematika, Tome 15 (2009) no. 7, pp. 137-140. http://geodesic.mathdoc.fr/item/FPM_2009_15_7_a5/

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[2] Herwig B., “Extending partial automorphisms on finite structures”, Combinatorica, 15 (1995), 365–371 | DOI | MR | Zbl

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