25 problems

Random-set Szemerédi conjecture. The $(k,\delta)$-Szemerédi property has threshold at $n^{-1/(k-1)}$ for random subsets of the integers. The case $k=3$ was already known to have th…

Let $P((0.5,1,0))$ denote the limiting proportion of MSTD subsets in the usual $(A,A)$ model. Martin–O'Bryant's conjecture. The proportion of MSTD sets, $P((0.5,1,0))$, is approxim…

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 $A\mathbf{x}=0$ be a system of homogeneous equations whose solution set is partition-regular, and let $p_*(n)$ denote the threshold for $[n]_p$ to contain monochromatic solutio…

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…

Selection conjecture. Each regularly varying random closed set admits a selection which is regularly varying on a suitably chosen nontrivial ideal.

Let $Q_d(p;m)$ and $\widehat Q_d(q;n)$ be the random sets appearing above, and let $A$ be a nonempty target set. In the critical case $2^dpq=1$, the relevant hitting probability is…

Let $D\subseteq [N]$ be $t$-intersective if every positive-density set $A\subseteq [N]$ contains a $(t+1)$-term arithmetic progression whose common difference lies in $D$. For the…

Let $T$ be the random Furstenberg set obtained from independent $(0,1)$-valued random variables $(\xi_k)_{k\geq1}$ with expectations $\mathbb{E}(\xi_k)=\delta_k$, using the special…

Let $k\geqslant 2$ and let $\beta_k$ be the supremum of all exponents $c<1$ such that, almost surely as $D\to\infty$, there are distinct sets…

Asymmetric random Rado threshold conjecture. The threshold for the property $[n]_p\to(A_1,\ldots,A_r)$ is $n^{-1/m(A_1,A_2)}$. This means that there are constants $0<c<C$ such that

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…

Let $P:[0,1]^3\to[0,1]$ be the limiting proportion of sum-dominant $(p,\rho_1,\rho_2)$-correlated pairs, and let $(A,A^c)$ denote the case $(\rho_1,\rho_2)=(0,1)$. Complementary-ca…

Mennink–Preneel conjecture. The quantity $\mu(A)$ should be close to $|A|$, up to possible logarithmic factors in $|G|$, for $\alpha\leq\frac12$, and should not much exceed…

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 $F_i$,

Random averaging conjecture. Then the random sequence is $\ell$-averaging.

Random intersectivity conjecture. Then the random sequence is $\ell$-intersective.

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.

Fix a constant $0<c<1/2$, and let $k_n$ be a sequence of integers satisfying … Let $[0,n]=\{0,1,\dots,n\}$, and consider subsets $S\subseteq[0,n]$ that are MSTD, meaning…

Rider-set nature conjecture. $p$-Rider sets with $p>\frac{4}{3}$ are not of the same nature as $p$-Rider sets with $p<\frac{4}{3}$.

For a finite set $A$, write $A+A$ for its sumset and $A-A$ for its difference set; call $A$ an MSTD set when $|A+A|>|A-A|$. Let a $P_m$-set be a set whose sumset and difference set…

Let $B\subset\mathbb{H}^2$ be a fixed open ball of radius $1$. Let $\mathcal Z\subset\mathbb{H}^2$ be an open random set whose law is invariant under isometries of $\mathbb{H}^2$,…

Let $I_N$ be the underlying finite interval, let $p:\mathbb{N}\to(0,1)$ satisfy the source's condition … f(x1,ldots,xk)=u1x1+cdots+ukxk, … thetaf={sigmain Sk:(u{sigma(1)},ldots,u{s…

Let $I_N$ be the underlying finite interval, and choose a random subset $A\subseteq I_N$ by including each element independently with probability $p(N)$. Call $A$ sum dominated whe…