An M-part Sperner theorem with applications to neural networks

Fundamental theorem of Sperner concerning the size of a family of sets unordered by set inclusion states that if the members of the family are subsets of an n-element set then the maximum size of the family is the largest binomial coefficient (_[n/2]ⁿ). Further, the families of size (_[n/2]ⁿ) must necessarily consist of: 1) all subsets of size n/2 if n is even, and 2) either all subsets of size (n-1)/2 or all subsets of size (n+1)/2 if n is odd. A generalization of this result is given that includes it as a special case. These results are applied to obtain a tight upper bound on the number of stationary points of Hopfield neural networks. A graph theoretic characterization of networks achieving this upper bound is also given