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I will present and discuss recent results and conjectures concerning the following open problem: Is it possible to define the affine dependencies over the reals of the set Cn:={0,1}n in a purely combinatorial way, i.e. without any reference to the field R? More precisely, I will sketch the deterministic approach to this problem via orientability of matroids proposed in [IS] and its relation with some recent probabilistic results. [IS] I.P.F. da Silva, Orientability of cubes, Discrete Math. 308 (2008), 3574-3585.
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