Question 4
Alice and Bob play a game. Bob starts by picking a set $S$ consisting of $M$ vectors of length $n$ with entries either $0$ or $1$. Alice picks a sequence of numbers $y_1\le y_2\le\cdots\le y_n$ from the interval $[0,1]$, and a choice of real numbers $x_1,x_2,\ldots,x_n\in\mathbb R$. Bob wins if he can pick a vector $(z_1,z_2,\ldots,z_n)\in S$ such that
$$\sum_{i=1}^n x_i y_i\le\sum_{i=1}^n x_i z_i,$$
otherwise Alice wins. Determine the minimum value of $M$ so that Bob can guarantee a win.
$$\sum_{i=1}^n x_i y_i\le\sum_{i=1}^n x_i z_i,$$
otherwise Alice wins. Determine the minimum value of $M$ so that Bob can guarantee a win.