Majority Decisions in Overlapping Committees and Asymptotic Size of Dichotomies

Abstract

In this paper we settle an open combinatorial conjecture in artificial neural networks: we show that the bound on the number of dichotomies given by Mitchinson and Durbin [Biological Cybernetics, 60 (1989), pp. 345--365] is tight, and that their structural asymptotics remain unchanged with varying required success probability. In our proof we use Rényi's graph-sieves inequalities, and we derive a contracted version and a sharp bound for a triple-indexed sum of binomial coefficients with dependent indices.