Threshold Logic Asymptotes

Let Rn^mbe the number of linearly separable n-argument functions specified on some m points of the n-cube. Let R̄n^mand Ṟn^mbe, respectively, the minimum and maximum values attained over all choices of the m points. It is known that R̄n^m<2mⁿ/n!. 1) Two published "simplifications" of the argument which establish this upper bound are shown to be fallacious. 2) It is proved that Ṟn^m≥4m^{(lg m−1)/2}. 3) It is proved that if m is less than exponential in n, then as n→∞, R̄n^m≈(m/n)^{n + lower order terms}. 4) Let Ln^mbe the maximum number of threshold gates needed to realize an arbitrary n-argument switching function specified on m points. It is shown that Ln^m≳2(m/lg m)^1/2.