8 problems

Let $[n]_p$ be the binomial random subset of $[n]$, obtained by retaining each integer independently with probability $p$, and call $[n]_p$ product-Schur if every $2$-colouring of…

Let $k$ be a positive integer and let $S\subset\mathbb{N}$ be chosen independently at random with … Here, $f=o(g)$ if and only if $g=\omega(f)$. Frantzikinakis–Lesigne–Weirdl conje…

Let $S\subset\mathbb{N}$ be chosen at random with $\mathbb{P}(d\in S)=\omega(1/d)$. A set of positive upper density means a set $A\subset\mathbb{N}$ for which…

Semirandom pointwise averaging conjecture. Then the random sequence $R^\omega=\{r_1<r_2<r_3<\dots\}$ satisfies, in every dynamical system, for bounded functions $F_i$,

Missing-sum monotonicity conjecture. For $m\geq m^*$, the following assertions hold:

Decreasing limiting-distribution conjecture. For all integers $m\geq 0$, this limiting function of $m$ is decreasing.

Random-set Szemerédi threshold conjecture. For every $\delta>0$ and every integer $k\geq 3$, there exist positive constants $c$ and $C$ such that

Fix a point $o\in\mathbb{H}^2$. For each $r>0$, let $p_r$ be the least $p\in[0,1]$ such that every random closed set $\mathcal Z\subset\mathbb{H}^2$ with isometry-invariant law and…