4 problems

The sharp stability conjecture. If $\mathcal{F}\subset\mathcal{P}([n])$ and $\mathcal{L}\subset\mathcal{P}([n])$ is as above, then there exists a family…

Let $n\geq 1$ and $t\geq 0$, let $\mu_t=\frac{1-e^{-t}}{2}\delta_{-1}+\frac{1+e^{-t}}{2}\delta_1$, and let $\mu=\mu_\infty$ be the uniform probability measure on $\{-1,1\}$. Write…

Let $A\subset\{0,1\}^n$ be monotone increasing, with measure $\alpha=|A|/2^n\leq 1/2$. Its edge-boundary is denoted by $\partial A$. Kahn and Kalai's conjecture. For every $L>0$, t…

Kahn–Kalai conjecture. For every $L>0$, there exist $L'>0$ and $\delta>0$ such that if