We define the notion of approximate Euler characteristic of definable
sets of a first order structure. We show that a structure admits a
non-trivial approximate Euler characteristic if it satisfies weak
pigeonhole principle WPHP2nn: two disjoint copies of a non-empty
definable set A cannot be definably embedded into A, and principle
CC of comparing cardinalities: for any two definable sets A, B
either A definably embeds in B or vice versa. Also, a structure
admitting a non-trivial approximate Euler characteristic must satisfy
WPHP2nn.
Further we show that a structure admits a non-trivial dimension
function on definable sets if and only if it satisfies weak pigeonhole
principle WPHPn2n: for no definable set A with more than one
element can A2 definably embed into A.
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