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.
"Approximate Euler characteristic, dimension, and weak pigeonhole principles." J. Symbolic Logic 69 (1) 201 - 214, March 2004. https://doi.org/10.2178/jsl/1080938837