5 problems

For every fixed integer $d\ge 1$, there exists a constant $C_d>0$ such that the following holds. If $F:\mathbb{R}^n\to\mathbb{R}$ is a degree-$d$ multilinear polynomial containing…

For every finite set $X$ and every hereditary family $\mathcal{F}\subseteq 2^X$—that is, $A\in\mathcal{F}$ and $B\subseteq A$ imply $B\in\mathcal{F}$—every intersecting subfamily…

For every sequence of families $\mathcal{A}_n\subseteq 2^{[n]}$ such that each $\mathcal{A}_n$ is increasing, regular, and $3$-wise intersecting, prove that…

With $\mu_a$ the biased-coin product probability measure on the product group $G$ defined in Section 3, Conjecture 6 states: Given $a>0$, there exist $C(a)>0$ such that for…

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