2 problems

Let $S_n$ be the symmetric group, and let $f\colon S_n\to\{0,1\}$ be a Boolean function. Say that $f$ is $\epsilon$-close to degree $d$ if its distance from the space of degree-$d$…

Let $S(n)$ be the symmetric group equipped with the metric $\mathrm{d}$, and let $A,B\subseteq S(n)$ be nonempty. Write $M$ for their midpoint set, and let $\mathrm{d}(A,B)$ denote…