Part 1 - University of Cambridge

30
Bijection Games
≡ Ck is also characterised by a k-pebble bijection game. (Hella 96).
The game is played on structures A and B with pebbles a 1 , . . .,a k on A and
b 1 , . . .,b k on B.
• Spoiler chooses a pair of pebbles a i and b i ;
• Duplicator chooses a bijection h : A → B such that for pebbles a j and
b j (j ≠ i), h(a j ) = b j ;
• Spoiler chooses a ∈ A and places a i on a and b i on h(a).
Duplicator loses if the partial map a i ↦→ b i is not a partial isomorphism.
Duplicator has a strategy to play forever if, and only if, A ≡ Ck B.
Anuj Dawar February 2013