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Approximate Euler characteristic, dimension, and weak pigeonhole principles

Published online by Cambridge University Press:  12 March 2014

Jan Krajíček
Affiliation:
Mathematical Institute, Academy of Sciences, Žitná 25, Prague 1, CZ-115 67, The Czech Republic, E-mail: krajicek@math.cas.cz
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Abstract

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 : 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 .

Further we show that a structure admits a non-trivial dimension function on definable sets if and only if it satisfies weak pigeonhole principle : for no definable set A with more than one element can A2 definably embed into A.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2004

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References

[1]Ajtai, M., On the existence of modulo p cardinality functions, Feasible mathematics II (Clote, P. and Remmel, J., editors), Birkhäuser, 1994, pp. 114.Google Scholar
[2]Ax, J., The elementary theory of finite fields, Annals of Mathematics, vol. 88 (1968), no. 2, pp. 239271.CrossRefGoogle Scholar
[3]Chatzidakis, Z., van den Dries, L., and Macintyre, A., Definable sets over finite fields, Journal für die reine und angewandte Mathematik, vol. 427 (1992), pp. 107135.Google Scholar
[4]Krajíček, J., Bounded arithmetic, propositional logic, and complexity theory, Encyclopedia of Mathematics and Its Applications, vol. 60, Cambridge University Press, 1995.CrossRefGoogle Scholar
[5]Krajíček, J., Uniform families of polynomial equations over a finite field and structures admitting an Euler characteristic of definable sets, Proceedings of the London Mathematical Society, vol. 81 (2000), no. 3, pp. 257284.CrossRefGoogle Scholar
[6]Krajíček, J., Combinatorics of first order structures and propositional proof systems, Archive for Mathematical Logic, (to appear, preprint 2001).Google Scholar
[7]Krajíček, J., Pudlák, P., and Woods, A., Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle, Random Structures and Algorithms, vol. 7 (1995), no. 1, pp. 1539.CrossRefGoogle Scholar
[8]Krajíček, J. and Scanlon, T., Combinatorics with definable sets: Euler characteristics and Grothendieck rings, The Bulletin of Symbolic Logic, vol. 6 (2001), no. 3, pp. 311330.CrossRefGoogle Scholar
[9]Paris, J. B., Wilkie, A. J., and Woods, A. R., Provability of the pigeonhole principle and the existence of infinitely many primes, this Journal, vol. 53 (1988), pp. 12351244.Google Scholar
[10]Pitassi, T., Beame, P., and Impagliazzo, R., Exponential lower bounds for the pigeonhole principle, Computational complexity, vol. 3 (1993), pp. 97308.CrossRefGoogle Scholar
[11]Schanuel, S., Negative sets have Euler characteristic and dimension, Category theory: Como '90 (Carboni, A, Pedicchio, M., and Rosolini, G., editors), Lecture Notes in Mathematics, vol. 1488, Springer-Verlag, 1991, pp. 379385.CrossRefGoogle Scholar
[12]van den Dries, L., Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, 1998.CrossRefGoogle Scholar

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