Geodesic
Lower bound for the number of inequalities representing a monotone Boolean function of $n$ variables
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 23 (1983) no. 3, pp. 754-756

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It is shown that there is a monotonic Boolean function whose representation by a system of linear inequalities with $n$ Boolean variables requires not less than $$\binom{n}{[n/2]}n^{-1}$$ inequalities. 
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     author = {Yu. A. Zuev and V. N. Trishin},
     title = {Lower bound for the number of inequalities representing a monotone {Boolean} function of $n$~variables},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     publisher = {mathdoc},
     volume = {23},
     number = {3},
     year = {1983},
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Yu. A. Zuev; V. N. Trishin. Lower bound for the number of inequalities representing a monotone Boolean function of $n$ variables. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 23 (1983) no. 3, pp. 754-756. http://geodesic.mathdoc.fr/item/ZVMMF_1983_23_3_a25/