3 problems

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…

A $3$-sunflower is a collection of three sets having the same pairwise intersections, and $[n]=\{1,\ldots,n\}$. Erdős–Szemerédi's Boolean-cube sunflower conjecture. There exists an…