Geodesic
On the number of threshold functions
Diskretnaya Matematika, Tome 5 (1993) no. 3, pp. 40-43

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A Boolean function is called a threshold function if its truth domain is a part of the $n$-cube cut off by some hyperplane. The number of threshold functions of $n$ variables $P(2,n)$ was estimated in [1, 2, 3]. Obtaining the lower bounds is a problem of special difficulty. Using a result of [4], Yu. A. Zuev showed [3] that for sufficiently large $n$ $$ P(2,n)>2^{n^2(1-10/\ln n)}. $$ In the present paper a new proof which gives a more precise lower bound of $P(2,n)$ is proposed, namely, it is proved that for sufficiently large $n$ $$ P(2,n)>2^{n^2(1-7/\ln n)}P\biggl(2,\biggl[\frac{7(n-1)\ln 2}{\ln(n-1)}\biggr]\biggr). $$ 
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     author = {A. A. Irmatov},
     title = {On the number of threshold functions},
     journal = {Diskretnaya Matematika},
     pages = {40--43},
     publisher = {mathdoc},
     volume = {5},
     number = {3},
     year = {1993},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1993_5_3_a2/}
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A. A. Irmatov. On the number of threshold functions. Diskretnaya Matematika, Tome 5 (1993) no. 3, pp. 40-43. http://geodesic.mathdoc.fr/item/DM_1993_5_3_a2/