6 problems

Let $p\text{-optimality}$ mean that a monotone Boolean function $f$ satisfies … For a set $J$ of coordinates, let $f_{J\to1}$ denote the function obtained by setting the coordinate…

Let $V\subseteq\mathbb{R}$ with $|V|=n$, and let the known distances be distributed as in $G(V,\lambda/n)$ for a parameter $\lambda>0$. Call a subset reconstructible if every injec…

Let $X$ be a sufficiently strong two-sided locally spectral expander, and let $f$ be a Boolean function on $X$. Write $I[f]$ for its total influence, and let $\tau$ be a face whose…

Friedgut–Kalai conjecture. There is an absolute constant $c$ such that, if $up^n(P)epsilon$, then

For the ring puzzle, let $p_{\epsilon}(n)$ be the unique value of $p$ such that the probability that the jigsaw process solves the puzzle is $f0a0\epsilon$, for fixed…

Let $S_n=[0,\sqrt{n}]\times[0,\sqrt{n}]\subset\mathbb{R}^2$, place points in $S_n$ according to a Poisson process of intensity $1$, and join each point to its $k$ nearest neighbour…