8 problems

Let $n$ be a positive integer and let $A$ be a subset of the Boolean cube with size $s=|A|$. Write $\Delta(s)$ for the minimum fractional edge boundary size among subsets of size…

Let $n$ be even and let $\mathcal{A}\subset\binom{[n]}{n/2}$ satisfy $\mu(\mathcal{A})=1/2$, where $\mu$ is the uniform measure on the middle layer, $\mathcal{A}^c$ is the compleme…

Let $n$ be even, let $\binom{[n]}{n/2}$ denote the middle layer of the Boolean cube, and let $\mathcal{A}\subset\binom{[n]}{n/2}$. Write $\mathcal{A}^c$ for its complement in this…

Abdi et al.'s conjecture. Non-trivial $r$-wise agreeing families cannot be cube-ideal for sufficiently large $r$.

Linear-factor witness conjecture. For $d\in[0,n]$ and $E\subseteq[0,n]$, if $j\in[0,n]$ and $j\notin\text{Z-cl}_{n,d}(E)$, then there exists a polynomial…

Generalized-monomial witness conjecture. For $d\in[0,n]$ and $E\subseteq[0,n]$, if $j\in[0,n]$ and $j\notin\text{Z-cl}_{n,d}(E)$, then there exists a polynomial…

Exact hyperplane cover conjecture. If $E\subsetneq[0,n]$ and $T_{n,2}\subseteq E$, then

Let $f$ be a harmonic multilinear polynomial, and let $f(\nu_{pn})$ and $f(\mu_p)$ denote its distributions under the uniform measure on the slice of the Boolean cube with paramete…