8 problems

Even-$n$, degree-two consequence. For every even $n$, $(n,2)$-HSFs do not exist.

Limiting Gotsman–Linial conjecture. The average sensitivity satisfies

Superweak Gotsman–Linial conjecture. For some function $g$ depending only on $d$,

Asymptotic Gotsman–Linial conjecture. There is an upper bound

Gotsman–Linial conjecture. For every $(n,d)$-PTF $f$,

Let $S\subseteq\mathbb{R}^d$ be a semi-algebraic set of description complexity $r$, and let $a_1,\dots,a_n\in\mathbb{R}^d$. Define $f:\\{-1,1\\}^n\to\\{0,1\\}$ by letting…

Let $T(n,d)$ be the number of $n$-variable polynomial threshold functions of degree $d$, and let … Here $\binom{n}{\le d}$ denotes the number of monomials in at most $d$ variables’…

Let $f:\{-1,1\}^n\to\{-1,1\}$ be a degree-$d$ polynomial threshold function, and let $\operatorname{\mathbf{I}}[f]$ denote its total influence. Weak Gotsman–Linial conjecture. For…