  We introduce a new technique proving formula size lower bounds based on the
linear programming bound originally introduced by Karchmer, Kushilevitz and
Nisan [11] and the theory of stable set polytope. We apply it to majority
functions and prove their formula size lower bounds improved from the classical
result of Khrapchenko [13]. Moreover, we introduce a notion of unbalanced
recursive ternary majority functions motivated by a decomposition theory of
monotone self-dual functions and give integrally matching upper and lower
bounds of their formula size. We also show monotone formula size lower bounds
of balanced recursive ternary majority functions improved from the quantum
adversary bound of Laplante, Lee and Szegedy [15].
