Suppose I want to have an integer program for handling the cases
$x_1>1\wedge x_2>1\wedge x_3>1\wedge\dots\wedge x_n>1\iff\delta=1$
$x_1>1\vee x_2>1\vee x_3>1\vee\dots\vee x_n>1\iff\delta=1$
how many number of integer variables are needed to handle case?
Is it possible at least one of them needs at most a constant number of binary variables?